| Page / Line |
Date |
Is / Ought |
|
vii / 18 |
(1/00) |
Replace the epitaph in the second dedication by:
So mayest thou, till suddenly, like a ripe fruit drop in thy mother's
lap. - from Paradise Lost by John Milton |
| xxiv
/ 5 |
|
/  |
|
/ 6 |
|
+
 /
= |
|
xxvii /-15 |
(4/00) |
(6.99) / (6.9) |
|
xxviii / 4 |
(4/00) |
R is locally / R is Goldie and locally |
| xxx
/ 7 |
(4/00) |
abetted, / abetted |
|
xxxvi / 15 |
(12/99) |
exists / exist |
|
xxxvii / 11 |
(12/99) |
transcendence / transcendence degree |
| xxviii
/ -2 |
(2/99) |
v / xxv |
|
xv /-23 |
(1/99) |
Indent
"Modules" |
|
xvii /-18 |
(1/99) |
Polynomals / Polynomial |
|
1 / 10 |
(12/99) |
I.
/ I. Rank need not be unique. See Cohn [77], pp 103-4. |
|
1 / 15 |
(1/99) |
In this case, / In case [S] = M then, |
|
1 / 17 |
(12/99) |
/  |
|
1 / 18 |
|
[76] / [72a] ( [72a] was actually published in '73) |
|
2 / 9 |
|
Index 2. / index 2. (5) The characteristic char R of a ring R is n
if there exists a least integer n > 0 such that n . 1 (= n sums
of 1) = 0. Then na = 0 VaeR. ... If no such n exists, then we say
char R = 0. |
|
4 /-10 |
(1/99) |
 /
every |
|
6 /-15 |
(2/99) |
that / that |
|
6 / |
(2/99) |
Add
Footnote 2: The term RD (= relatively divisible) was used by Warfield
[69a]. See Fuchs, Salce, and Zanardo [99] for a
historical sketch. Cf. 6.46A. |
| 9
/ 20 |
(2/99) |
for a prime p. / for a prime p. Then f has characteristic 0 in the
first instance, and p in the second. Also see Kleiner [99] for
a sketch of the development of the axioms of fields, including the
contributions of Weber, Dedekind, Hensel, Steinitz, Artin and
Schreier, among others. |
|
9 /-3; 10 / 5 |
(12/99) |
algebraicaly / algebraically |
|
15 / 9 |
(2/99) |
ideal. / ideal. Conversely, every simple right R-module V is cyclic,
in fact, V = vR for any nonzero element v in V. Moreover V is
isomophic to R/I, where I is the annihilator of v in R, hence a maximal
right ideal. |
|
16 / 12 |
(12/99) |
[91]
/ [95] |
|
17 /-11 |
(3/99) |
sup.2.17s8./ sup. 2.17A and 2.17F (pp. 31 and 33). |
|
18 / 1 |
(1/99) |
algebra / central algebra |
|
21 /-12 |
(12/99) |
Add: k[[x]] also denotes the power series. |
|
24 / 4 |
(3/99) |
P.M. Cohn / P.M. Cohn [61] |
|
25 / 18,19 |
(2/99) |
of a subfield / of a proper subfield |
|
27 /-12 |
(5/99) |
  
/ ' |
|
31 / 2 |
|
of a set / of subsets of a set |
|
31 / 5 |
|
that
M / that a module M |
|
31 / 20 |
(12/99) |
Delete
S 0 |
|
31 / 21 |
(12/99) |
Add: Note that S is nonempty. |
| 32
/ 7 |
(12/99) |
irreducible / irreducible qua submodule, |
|
33 / 17, 18 |
|
Delete the entire offending clause beginning with "although" and ending
with "cyclic." |
|
34 / 13,14 |
(12/99) |
Preceding the paragraph heading "Hilbert Basis Theorem," add:
2.19C COROLLARY. If R is commutative Noetherian but non-Artinian ring,
then there exists an ideal I maximal with respect to the property
that R/I is non-Artinian. Futhermore, any such ideal I is a prime
ideal and R/K is Artinian for any ideal K properly containing I. Proof
Exercise. |
| 35/-11 |
(10/00) |
number
field L./number field L. See, e.g., Osofsky [99] for an elementary
proof. |
|
35 /-1 |
(12/99) |
Add to the footnote: P.M. Cohn points out that the "usual spelling
is Shafarevich." |
|
36 / 2 |
(12/99) |
iteralia
/ inter alia |
|
38 / 10 |
(12/99) |
co-irreducible / irreducible (formerly called co-irreducible) |
| 39
/ 1 |
(12/99) |
[Z-W]
/ [ Z-S] |
|
43 / 16 |
(10/99) |
P
/ B |
|
52 / |
(12/99) |
In 3.2D: Schöpf / Schopf |
| 53
/ 4 |
(12/99) |
1-1 / injective |
|
55 / 1 |
(4/00) |
THEOREM / THEOREM [60] |
|
55 /-11 |
(10/99) |
cogenerator / generator |
|
56 / 10 |
(3/99) |
Theorem 3.8 / Theorem 3.7C |
| 56/
17 |
(12/99) |
R. / R.} |
|
57 / 5 |
|
 (, )
/ ( ,) |
|
60 / -15 |
|
-injective
/ -injective (Cf.3.7A) |
|
61 / 12 |
(4/00) |
Goursaud and Valette / Goursaud and Valette [75], who prove that any
ring R with a faithful sigma-injective right R-module has acc on direct
sums of right ideals. See finite Goldie dimension below. |
|
61 / -2 |
(2/99) |
cf.
Chapter 8 / cf. Chapter 8 and 16.9B. |
|
62 / 1 |
(2/99) |
in
a variable x / in any finite or infinite number of variables |
|
62 / 6,7 |
(4/00) |
To "REMARKS" add: (4) For some pathology on the Goldie dimension of
a sum of two submodules, see Camillo [78b], Camillo and Zelmanowitz
[78], and Valle [94]. |
|
62 / 12 |
(2/99) |
rings. / rings. Shock 72, Theorem 3.6, extended Small's theorem to
infinite polynomial rings. |
| 62
/ 15 |
(4/00) |
Add: For theorem 3.13 for rings with involution, see Domokus [94]. |
| 64
/ 1 |
(12 /99) |
An R-module / An injective R-module |
|
64 / 15 |
|
Cf.4.6D / Cf. 7.32s |
| 64
/ |
(2/99) |
Following 3.15C, add: Remark. Matlis [58] proved that any injective
module E over a Noetherian commutative ring R had this structure and
that there is a 1-1 correspondence between prime ideals P and indecomposable
injectives E(R/P). Cf. Theorem 3.4 above. Also see Goodearl-Warfield
[89], Theorem 4.24, p. 79. |
|
64 / 17 |
|
/  |
|
64 / |
(2/99) |
To Theorem 3.16C add: 4. R has Noetherian quotient ring R REMARK 4.
is Beck's Corollary 3.10 We come back to these ideas in 16.33. |
|
64 / |
(3/99) |
Preceding
Theorem 3.17B, add: REMARK. Note that requiring the acc on annhilators
in R/I suffices for (3) of 3.17A. Cf. Mori domains, 9.4s, p. 164. |
|
65 / |
(12/99) |
In
2. of 3.19A: equivalent / equivalently |
| 66
/ |
(12/99) |
In 3.24A, delete "f.g" |
| 66/1 |
(6/00) |
Cozzens
[70] / Cozzens [70] and Koifman [70] |
|
66/6 |
(6/00) |
Resco
[87]. / Resco [87]. See Komarnitskii [97] for solution
to a question stated in the title ibid. |
| 67
/ 8 |
|
ideal
/ proper ideal |
|
67 / 17 |
|
free.
/ free. Cf.3.23C |
|
68 / 1,3 |
(1/99) |
lift/ring should be lift/rad ring |
|
68 / 2,3 |
(1/99) |
See Jacobson...SBI-rings / Jacobson's SBI-rings in his book [56,64],
p.53, are lift/rad rings. |
| 68
/ 12 |
(12/99) |
modules / module |
|
70 / -18 |
|
3.33F R is / 3.33F. R is |
|
70 /-2 |
(12/99) |
Sätzes
/ Sätze |
| 74/-14 |
(12/99) |
add: Graham Higman [56] proved it in greater generality. However,
in his review (MR, 92d: 13023) of Formanek's Survey [90],
E. Zel'manov states (without a reference!) that J.Dubnov and V. Ivanov
proved the Nagata-Higman Theorem in 1943. |
|
75 / 5 |
(2/99) |
See / see / 16 (1/99) nilalgebra / nil algebra |
|
77 / -6 |
|
Similar
Notation / Similar. Notation |
|
81 / 10 |
|
of M. / of M onto M. |
| 84
/ |
(2/99) |
Above
Kolchin's Theorem, add: Remark. See Shock [72b] for generalizations
to modules M of finite Goldie dimension or acc on "rationally closed"
submodules (see 12.OB). In the latter case, every nil subring
S of A is nilpotent. This generalizes a theorem of Small for Noetherian
M. (See op.cit., p.313.) |
|
/ -18 |
|
If g is / If t is |
|
88 / |
(2/99) |
Add: 3.81 Theorem (Ibid.) If R is commutative, then prim rad R is
essentially nilpotent |
|
89 / 8 |
(12/99) |
1 + i / VNR(i) |
|
89 / 14 |
(12/99) |
is
/ if |
| 91
/-13,-12 |
(2/99) |
C.V.
Jensen / C.U. Jensen |
| 94/
-1 |
)2/01) |
generalizes/
is 4.5A Thereom (Kaplansky [50]) Let R be a VNR ring, and n and interger
>0. Then R ~An for
an Abelian VNT ring A iff all (left) primitive factor rings have index
n. |
| 95/
8 |
(2/01) |
Kaplansky
[50] (all R/M have the same index n and / Kaplansky [50] without assuming
self-injectivity when all R/M have the same index, and then |
|
95 / 11 |
(4/00) |
Utumi. / Utumi. Also see Carson [76] proves 4.5 for complete VNR maximal
left quotient rings of finite
index. |
|
95 /-10 |
(12/99) |
Was Sind und Was Sollen Die Zahlen / Was sind und was sollen die Zahlen |
|
100 /-11,-12 |
(3/00) |
Add preceding 4.16C: REMARK. Sweedler [75] (for commutative algebras)
and Lawrence [76] characterize when a tensor product of algebras is
local |
|
101 / 10 |
|
4.26C / 4.2C |
|
/ 18 |
(3/99) |
-ring /
 -ring |
|
102 / 8 |
(11/99) |
13.7 / 13.7. Also see Rutter [71], where this theorem is extended
to the endomorphism ring of "PF-modules." |
| 104
/ 14-15 |
|
Following line 14, preceding Remark add: 4.32 Theorem (Kitamura[91]).
If A is a separable k-algebra whose center is a free k-module, then
A is FPF iff k is FPF. Note. This extends results of S. Page for when
C=k, and Herbera - Menal [89] for the group algebra A = kG of a finite
group of unit order. |
|
104 / 16 |
|
Page[84e] / Page [84] |
|
106 / 13 |
(6/99) |
113.7A / 13.7A |
|
108/-9 |
(4/00) |
(See
5.4B)./ (see 5/4B). Furthermore:
5.4 (Theorem
(Nagata [62], p. 55, 17.7) Let R be a Noetherian commutative semilocal
ring with Jacobson radical m, and let ( ,... )
be an ideal whose radical is m. Then the completion R* of R in the
m-adic topology is isomorphic to the power series ring A in n variables
over R modulo the ideal (
-  ,
...,
-  )
of A.
5.4
Theorem (Chevalley [43]) Let  ,
...
be the maximal ideals of R in Theorme 5.4 .
Then R* is the finite product of the completions of the local rings
of R at  , =1,...,n. |
|
109 /-3 |
(3/00) |
VÁMOS
[77,79] / VÁMOS [77b,79] |
|
110 / 4 |
(3/00) |
[76] / [77b] |
|
111 / 14, 15 |
(12/99) |
Add: 5.14C Remark. According to Kleiner [99], the concept of a valuation
of general fields extending Hensel's p-adic valuations is due to Kürshak
in 1913, who proved the existence of their completions, and in 1918
Ostrowski determined all valuations of the field Q of rational numbers. |
|
111 /-15,-16 |
(12/99) |
Maclane
/ Mac Lane |
|
111 /-2,-3 |
(4/00) |
Also see F.K. Schmidt [33] and Schilling [50], Chap. 7, on multiply
complete fields. |
| 112
/ -5 |
|
requires a theorem / requires ideas of a theorem / -5 See 7.21ff /
See 7.21ff and "Letter from Victor Camillo", p. 254. |
|
118 / 3 |
|
Let R be semiperfect / Let R be a semiperfect ring |
|
118 /-13 |
(3/99) |
Faith-Page [81] / Faith-Page [84] |
| 119
/ 3 |
|
Faith-Page [81] / Faith-Page [84] |
|
/ 7 |
(2/99) |
Remark 4.84f / Example 4.24 |
| 120
/ 5 |
(2/99) |
VD). / VD). Also see 4.24-5. |
|
121 /-1 |
(2/99) |
Proposition, p. 84) / Proposition, p. 84. In fact, pure-injectivity
can be viewed as injectivity in an appropriate category; see Gruson-Jensen
[73].) |
|
123 / 1 |
(2/99) |
50 / some 40 |
|
/-10 |
(2/99) |
Hendriksen / Henriksen |
|
124 / |
(2/99) |
Preceding 6.3F, insert: Remarks. (1) Any ring with stable range is
Dedekind finite (if xa = 1 then, taking b = 0 in the definition, we
see that a is a unit.) (2) The above definition for stable range 1
is right- left symmetric, as T.Y. Lam points out in his Exercises
in Classical Ring Theory (Springer Verlag) pp. 15-16, Ex. 1.25 and
comment. (I have Lam [99b] to thank for these remarks.) |
|
124 / -1 |
|
Faith
[91b] / Faith [91b]. Cf. Theorem 16.31. |
| 125
/ 1 |
(1/99) |
A ring R / A commutative ring R |
| 126/
-7 |
|
(IF4),
R is then / any VNR ring R is |
| 127/ |
(11/00) |
Insert
preceeding REMARK 6.14A Theorem. Let R be strongly regular.
(1) (Hirano, Hung and Kim [95]) If R is &o-injective,
then R<x> C =R[[x]]) is a duo ring and Bezont.
(2) (Karamzadeh and Koochakkapoor [99].) The converse of (1) holds. |
| 128/
-5 |
|
in
A. R / in R. R |
|
128/19,20 |
(9/00) |
Following "REMARK" add:6.19A Theorem. (Brandal [73],
Olberding [99]) Every homomorphic image of the quotient
field Q of a domain R is injective iff R is an AMVR Prüfer domain.
REMARK. See loc.cit. for other results, and see the Math. Rev.
(2000f: 13041) of Olberding's paper by M. Fontana for historical background. |
|
129 /-9,-10 |
(4/00) |
Add: Cf. R.E. Johnson [69] for other examples of Mal'cev domains. |
|
129 /-8 |
(6/99) |
field / sfield |
|
130 |
(12/99) |
Above 6.31, insert: Definition. A ring R is a right (semi) fir if
every (f.g) right ideal is free of unique rank. |
|
131 / 18 |
(12/99) |
contructions / constructions |
| 132
/ 9 |
(12/99) |
injective / quasi-injective |
|
132 / 10 |
(2/99) |
(ibid.) / (ibid.) Also, Kawada [57] showed that a right self-pseudo-injective
algebra of finite dimension over an algebraically closed field is
QF. |
|
133/ 6 |
(3/99) |
Q(R[X] / Q(R[X]) |
|
/ 10 |
(11/99) |
intersection, then / intersection. Then |
|
/ 11 |
|
Cf. sup. 7.8 / Cf. 3.58-61 and sup. 7.8 |
|
133 |
|
preceding
"6.39 Theorem", add this paragraph:
ASSOCIATED
PRIMES OF POLYNOMIAL RINGS
For
a commutative ring R, an associated prime ideal is a prime ideal
P that is the annihilator of some element of R, and Ass R denotes
the set of all such P. (See 16.11). By theorem 2.37E, any maximal
annihilator ideal T is an associated prime ideal (also see 16.12),
and Ass*R denotes the set of all
such T. |
| |
(9/99) |
Replace "6.39 Theorem and Remark" by:
6.39 Theorem (Brewer-Heinzer [74]). For any commutative ring R, the
extension mapping of Ass R to Ass R[X] sending P to P[X] is a bijection.
Thus: card Ass R = card Ass R[X] and card Ass*R = Ass* R[X]. Remark.
The proof employs the concepts of Lazard [69], and localization theory.
The author [00b] has given a direct proof using results of Shock [72b]. |
| 133
/-12 |
(12/99) |
McCoy[57] / McCoy [57b] |
| 135
/ 7 |
(2/99) |
[98] / [99] |
|
/ 9 |
(2/99) |
(DVD).
/ (DVD). This is a corollary to ibid. Theorem 6 that states that "RD-essential"
is transitive over a domain R iff R is a DVD. Cf. 6.46A. |
|
136 |
(12/99) |
Trilfaj / Trlifaj |
|
136 /-1 |
(12/99) |
1.24B / 1.24B Cf. also Azumaya [96] who points out that if every projective
left module is pure-injective, then R must be left perfect. |
| 137
/ 9 |
|
direct summands. / direct summands. Cf.8.6s |
|
139 / 5 |
|
restricted Artinian / restricted Artinian. Cf. 2.19B. |
|
140 /-1 |
(2/99) |
Add: Cf. 13.45(3) |
|
140 |
(4/00) |
Following
Theorem 7.7 add:
7.7A THEOREM (Cozzens and Faith [75], p. 138) Any right ideal of a
simple right hereditary ring can be generated by 2 elements. 7.7B
REMARK. See Ibid, p. 33, Lemma 2.24 for a short proof of the first
statement of Theorem 7.7. Also see theorem 7.12 below. |
|
140 /-3 |
(12/99) |
Sandomerski / Sandomierski |
|
141 / 5,6 |
(12/99) |
Add: 7.8A THEOREM (Jategaonkar [74b] Over a (twosided) fully bounded
Noetherian ring R any f.e. module M is Artinian, and if M is also
f.g, then M has finite length. Remark. For a commutative Noetherian
ring this is a theorem of Matlis [58]. |
| 141 |
(4/00) |
Add to Theorem 7.12: Any right Goldie simple ring R of gl.dim not
exceeding 2 is right Noetherian (Ibid., Theorem 2.25, Proposition
2.38 and Theorem 2.40.) |
| 145
/ -9 |
|
Add: By Theorem 3.33A, M is a min module iff semiArtinian. |
|
146 / 1,2 |
(2/99) |
Shock's Theorem...latter. / Cf. Vic Camillo's Letters, p. 254. |
|
148 / -9 |
|
Cf. 13.62(2) / Cf. Shock's Theorem 7.28 |
|
150 /-7 |
(10/99) |
Delete the previous addendum of (3/99). |
|
/ -6,-7 |
(10/99) |
Insert: Theorem. An injective right R-module E is
-injective iff E is countermodule of finite length.
Proof. This follows easily from Prop. 3.8(a), and the Teply-Miller
Theorem 3.10. Cf. p. 30 of the author's lectures [82a], esp. Corollary
7.5. |
| 153/14,15 |
(11/00) |
Krull
[28] - Schmidt [25,26] / Krull [25,26] - Schmidt [28] |
| 156
/13 |
(1/99) |
A ring R...provided that / A ring R is a lift/rad ring provided that |
|
/ 18 |
(1/99) |
R
is SBI / lift/rad ring |
|
157 / 16 |
(2/99) |
Harano
/ Hirano |
|
/ -18 |
|
indempotents / idempotents |
|
/ -10 |
|
isormophic / isomorphic |
|
/ -23 |
(1/99) |
Replace Example 4 by : Example 4 (Lam). A Dedekind finite exchange
ring R need not be semiperfect, e.g., any infinite product R of copies
of any field: R is self-injective hence suitable by 4.2A hence an
exchange ring by 8.4C. |
|
158 |
(2/99) |
To Corollary 8.5B, add: (Cf. 8.1s and 8.3.) |
| 158
/-12 |
(4/99) |
esp. 16.39 / esp. 16.39-16.43 |
|
159 /-6 |
(4/00) |
ring; / ring A; |
|
159 /-5 |
(4/00) |
R-module / R-module M |
|
159 /-1 |
(4/00) |
M and N. / M and N. A necessary condition for the truth of the conjecture
is that the center of A (= the biendomorphism ring of M) be linearly
compact. |
|
163 / 5 |
(2/99) |
Faith [86] / Faith [86b] |
|
/ 8 |
(2/99) |
Faith
[86] / Faith [89a] |
|
163 /-3,-4 |
(4/00) |
Add: REMARK. Kerr [79] gives an example of a commutative Goldie Kerr
ring whose 2x2 matrix ring is not Goldie. (In this example R has just
two nontrivial annihilator ideals and Goldie dimension two!
Cf. Theorem 9.4 & Corollary 9.5.) |
|
163 /-3 |
(3/00) |
Roitman [89,I] / Roitman [90] |
| 164
/ 3,4 |
(3/00) |
Insert: REMARK. Roitman applied the coup de grace to Theorem 9.3 in
the following: 9.3' THEOREM. Over a countable field k there is an
acc  algebra that
is not Kerr. |
|
164 /12 |
(1/99) |
[93,94,96] / [91a, 94,96b] |
|
164 / 16,17 |
(4/00) |
Add: REMARK. Since Q is Artinian in Theorem 9.4 (and the next corollary),
then any n x n matrix ring over R satisfies acc ,
hence is Goldie. |
|
164
|
(1/99) |
There
is a gap in the proof of Theorem 9.6: In order to invoke Beck's Theorem,
Ass R must be finite. The set Ass*R of maximal associated primes is
finite in a ring with acc
by Theorems 16.25 or 16.31. Is Ass R finite in this case? Since Ass
R is finite in a ring of finite Goldie dimension by Corollary 16.18,
then "Theorem 9.6" holds assuming, e.g., that R is Goldie.
Hence replace 9.6 by:
9.6 Theorem. If R is a commutative Goldie ring, and if the local ring
of R at each associated prime ideal P is Noetherian, then R has a
flat embedding in a Noetherian ring, hence R is Kerr, and so is any
polynomial ring R[X] over R.
Proof. See Beck's Theorem 3.16C. Also see Theorem 16.33 and Remark
16.34. |
|
165 /-16 |
(2/99) |
[84b,c]
/ [84b, 86c] |
|
/-14 |
(2/99) |
[84c] / [86c], p. 40, theorem 2.1 |
|
/- 13 |
(2/00) |
Roitman [94,II] / Roitman [89,II] /-11 (12/99) element of K / element
x of K |
| 166
/ 12 |
(12/99) |
integral / integrally |
|
166 /-15 |
(7/99) |
conversely if K / conversely if A |
|
167 / 6 |
(2/99) |
9.16B / 9.15B |
|
168 / 16 |
(6/00) |
domains. / domains. Also see Osmanagic [99] for a general approximation
theorem for "non-Marot Krullrings." |
|
170 /13 |
(10/99) |
/  |
| 171/-9,-8 |
(1/99) |
Put
"generalized discrete valuation domain" into bold face type. |
|
/ -7 |
(2/99) |
M / m |
| 174
/ 9 |
(2/99) |
is isomorphic to S / R is isomorphic to S |
|
174/-5,-4,-3 |
(1/99) |
(1), (2), (3) / (4), (5), (6) |
|
175 / 9 |
(12/99) |
finite. / finite (and then A is right SI by 11.1.) |
|
177 / 15 |
(12/99) |
Higman [56] / D.G. Higman [54] |
|
178 / 3 |
(3/99) |
domain? / domain when K is ? |
|
179 / 10,11 |
(2/99) |
these rings...defined / these rings R are essential R-submodules |
|
180 -21 |
(2/99) |
large
/ essential |
|
181 / -15 |
|
Right singular / Right nonsingular |
|
182/-8 |
(1/99) |
z(FG) / Z(FG) |
|
183/18 |
(6/00) |
6.30).
/ 6.30). Also see Kharchenko [00] for a survey. |
|
184/-21 |
(1/99) |
Gabriel
/ P. Gabriel |
| 186
/ 19 |
|
complement
of  /complement
of  |
|
186/-10 |
(1/99) |
module / modular |
|
187 / 6 |
(2/99) |
It
can be...that / As stated in Theorem 12.4 |
|
189 /-18,-19 |
(12/99) |
Zimmerman / Zimmermann |
|
189 /-9 |
(12/99) |
Chatters and Hajarnavis [77]. / Chatters and Hajarnavis [77], who
explicitly characterize indecomposable right CS rings R which are
either semiprimary or have semi-primary right quotient rings as certain
triangular matrices over subrings of division rings. The case when
R is right Artinian is then classified (Theorems 3.1 and 6.10.)
Furthermore:
12.5A. Theorems (Chatters and Hajarnavis [77],
(1) A twosided Noetherian CS ring R is a pp ring with twosided maximal
quotient ring.
(2) Any twosided Noetherian CS ring has a CS Artinian quotient ring
(Corollary to Theorem 6.5; Cf. Example 6.6, loc.cit.)
(3) A right Noetherian right nonsingular right CS ring has Artinian
quotient ring (Prop. 6.7. loc.cit.)
(4) An indecomposable twosided Noetherian right nonsingular right
CS ring is either prime or Artinian.
(5) A twosided Noetherian prime ring R is twosided CS iff R is a
pp ring. (Theorem 6.8; Cf. Example 6.9, loc.cit.)
(6) A left Noetherian right PIR is twosided CS. (Corollary to Theorem
6.8, loc.cit.) Remark. The right CS property is not a Morita invariant
property: there is a full 2x2 matrix ring over a right Noetherian
right hereditary domain D that is not left Ore is not right CS.(Example
6.9, loc.cit.) |
| 190
/ 20 |
(11/99) |
Replace
4. by the following: 4. If S is a submodule of a f.g projective right
R-module P that is minimal with respect to S + T = P for some submodule
T, then S (called a "complement submodule,"ibid.) is a direct summand
of P. |
|
/ 25 |
(11/99) |
the
next theorem. / proofs of Theorem 12.9 and the next theorem. |
| 190/-4,-2,-1 |
(1/99) |
AFT / ATF |
|
191 /-4 |
(12/99) |
overring / any overring |
|
194 / -13 |
|
Corollaries
10.14 and 10.15. / Corollaries 10.14 and 10.15. Also see Theorem 7.45. |
| 195/
-19 |
(1/01) |
2.24.6B
/ 25.4.6B |
| 195
/-9 |
(4/00) |
finitely
embedded / finitely embedded linearly compact |
|
195 /-2 |
(2/99) |
ring
/ Noetherian ring |
|
196 /-12 |
(2/99) |
Cf.
13.19 / Cf. 13.19 and 13.36 |
| 197
/ 2 |
(2/99) |
7.21ff. / 7.27ff. |
|
197 / 7 |
(12/99) |
Leptin [14] / Leptin [55,57] |
| 197 |
(1/99) |
To "REMARK" add: 6. Any AB-5* right R-module M over a right l.c. ring
R is l.c. |
|
197 |
(1/99) |
To "Note" add: 4. (6), and other AB-5* results, are by the same authors
[98]. |
|
198/10 |
(1/99) |
3.18 / 13.18 |
|
198 /13 |
(1/99) |
[63] / [63b] |
|
199 / 3 |
(2/99) |
13.22.4 / 13.22.4 and 13.24A |
| 200/
-19 |
(1/01) |
Theorem
3.5C / Theorem 3.5D |
|
200 /-17,-18 |
(2/00) |
Add: 13.30 THEOREM (Jain [73]) Any right FGF ring R is right IF. PROOF.
This is a corollary of Theorem 6.8. REMARK. Jain proved this in a
different way. |
|
201 / 3 |
(2/00) |
FPF / FGF 201 |
|
/ 6,7 |
(3/00) |
Add: REMARK. Any right coherent right FGF ring R is QF, since every
cyclic right R-module R/I embeds in a free module of finite rank,
hence is finitely presented, i.e., I is f.g. Thus R is Noetherian
so (FGF2) applies. |
|
201 |
(2/99) |
In
13.32-3: Menal [82] / Menal [82b] |
|
201 /-10 |
(1/00) |
Pardo / Pardo and Asensio |
| 201
/-9 |
(12/99) |
[97A] / [97] |
|
202 /1 |
(2/99) |
[83] / [82b] |
|
203 / 11 |
(4/99) |
Theorem 3.31 / Theorem 13.31 |
|
203 /-1 |
(2/99) |
Add: Regarding (3), cf. Theorem 7.7. |
|
206 /-5 |
(4/00) |
8.12 / Theorem 3.4A |
|
207 / 5 |
(11/99) |
[69a] / [69b] |
|
20 /-15 |
(2/99) |
dim  (m/ )
/ dim (m/) |
| 210
/3,4 |
(4/00) |
Add:
14/16A COHEM'S STRUCTURE THEOREM [46]. Let (R,m) be a complete regular
local ring of dimension n. If R has equicharacteristic (i.e., char
R = R/m), then R is isomorphic to the power series ting over R/m
in n variables. (See Corollary 14.19 below. Also see Theorem 5.4.)
Proof. See, e.g. Zariski-Samuel [60],p. 307 |
|
210 / 8 |
(12/99) |
exists / exist |
|
210 /-18,-19 |
(2/00) |
Add: COHERENT POLYNOMIAL RINGS
By a result of Soublin [68b] a polynomial ring over a coherent commutative
ring R need not be coherent, but it is if R is VNR (Soublin [68a]).
Cf. Carson [72]. Also see Theorem 14.21 below. |
|
210 /-3,-4 |
(2/00) |
Add: REMARK. A locally coherent commutative ring R need not be coherent,
but it is if R is semilocal (Harris [67]). Furthermore: (4/00) 14.21A
THEOREM (Carson [78]) If R is a VNR ring of bounded index, the R[X]
is coherent for any set X of commuting variables. |
|
212 / 18 |
(2/99) |
See (loc.cit.) / See Gordon-Robson (loc.cit.) |
| 213
/ 14 |
(2/99) |
dim R< .
/ dim R<.
Cf 14.25, also 14.31A below. |
|
215 |
(2/99) |
Following 14.36, add: Remark. The same result holds replacing "Radical"
by "prime". See Corollary 2.4 op.cit. |
|
215 / 19 |
(12/99) |
Pusat-Yilmaz,
and Smith [96] / Kaplansky [74] |
|
215 |
(2/99) |
Following 14.38 add: Remark. The necessity of the conditions is, of
course, 14.34; and the sufficiency is Exercise 25, p. 65 of Kaplansky
[74]. Moreover Theorem 14.38 holds more generally for PI - rings (Pusat-Yilmaz
and Smith [96]) |
| 217
/ -10 |
(2/99) |
Countable ordinal / ordinal |
|
/ -9 |
(2/99) |
ring
/ unique factorization domain. (Loc.cit. Corollary 9.11) |
|
219 |
(9/99) |
Add:
FACCHINI'S THEOREMS ON INJECTIVE DIMENSION
Facchini [82] studied rings R with the property
(P) that finitely embedded right R-modules M have inj.dim. not exceeding
1; equivalently, every factor module of E(M) is injective. (It suffices
that this is required of just simple modules M. (Lemma 1.4).)
Facchini's main theorem characterizes commutative
rings R with the property (P) as locally almost maximal valuation
rings whose prime ideals are either minimal or contained in a unique
maximal ideal (Theorem 2.31). In case R is semilocal, then R is a
direct pro- duct of almost maximal Bezout domains, and conversely,
(Corollary 2.4). Moreover, R has the property (P) iff R is reduced
and E(R/P) is uniserial for any non-minimal prime ideal P (Theorem
3.1). |
|
222 / 7 |
(12/99) |
Corollary 5.3 / Proposition 9.5.3 |
|
223 /-9,-10 |
(2/99) |
no
coefficient...annihilates A. / when A does not have a unit element,
assume that not all coefficients of f annihilate A. (See strongly
regular, 15.9 and Definition 1' on p. 225.) |
|
225 / 18 |
(12/99) |
Cohn [8l] / Cohn [91] 226 / 1,4 (4/99) Change the subscript on S^m
to 2n. |
| 226 |
(4/99) |
 /  |
|
226 / 8 |
(2/99) |
algebra k / algebra C over k 229 To "Remark 16.2", add: Cf.3.7B and
C. |
|
230 / 1 |
(12/99) |
Capson / Copson |
/-1
|
(12/99) |
x / (x) |
|
231 / 15 |
(12/99) |
right ideal of R. / right ideal of R (cf.16.9C below). |
| 232
/-13 |
(12/99) |
theorem
/ lemma |
|
233 / 12 |
|
Add:
Ass*( )
= Ass(M) / Ass*()
= Ass*(M), in fact, the set of associated primes of a finite product
of modules is the union of their associate primes. (See, e.g., Eisenbud
[96], p. 93, Lemma 3.6, for a more general result.) |
|
233 / 21 |
(9/99) |
Add: Cf. the Brewer-Heinzer Theorem 6.39. |
|
235 / 9 |
|
Indedecomposable / indecomposable |
|
235 |
|
16.17 Theorem is worded awkwardly.
Replace by:
16.17 Theorem. If M is a module over a commutative ring R, and if
E(M) is directly decomposable, then d(M)>! Ass(M)!. |
|
/ 17 |
|
Delete the first sentence of the proof, and delete "In the second
statement," |
|
235 / 18 |
|
we may / We may above remark. / above definition. |
|
238 |
(2/00) |
Preceding 16.33: replace the definition by :
Definition. Let
O -> M ->  ->
... ->  ->
...
be a minimal injective resolution of the R-module M, and define the
Noetherian depth of M, denoted n.d.M as the maximal i such that  is
sigma-injective V n< i. If
is not sigma-injective, we let n.d.M = -1; and if
is sigma-injective for all i, set n.d.M = . |
|
238 /-4 |
(3/99) |
Beck [72] / Beck [72A] |
|
239 |
(2/99) |
Preceding 16.34 insert: 16.33A BECK'S COROLLARY 3.10. A commutative
ring R has n.d. >0 iff
Q(R) is Noetherian. |
| |
(10/99) |
to 16.34 add: (4) 16.33(1) is equivalent to Q(R) being Noetherian.
Thus by Camillo's theorem 16.50 (coming up), every factor ring of
R satisfies 16.33(1) iff R is Noetherian. |
|
239 / 8 |
|
zip.
/ zip. (3) Cf. 3.16C. |
|
/ 21 |
|
Also dee / Also see |
|
240 / -17 |
|
R/P / R \ P |
|
241 /-12 |
(2/99) |
1.10 / 16.48 |
|
242 / -9 |
|
Lam[98a] / Lam [98b] |
|
/ -4 |
|
Delete question (3). |
|
246 /-20 |
(12/99) |
implications. there / implications; there |
|
248 / 18 |
(2/99) |
4.1A
applies. / 17.4A applies |
|
254 / 8 |
(12/99) |
Small [86] / Small [85] |
|
254 /-5 |
(12/99) |
superceded / superseded |
| 254
/14 |
(5/00) |
independent
/ dependent |
|
255 / 3 |
(1/99) |
1950 /1951 |
|
/ 12 |
(2/99) |
about this.) / about this. Also see Bashmakova and Smirnova [99] (Amer.Math.Monthly
106 (1999))
for an attribution of "literal symbolism" of algebra to Diophantus
(middle 3rd Century B.C.) |
|
255 /-10 |
(1/00) |
name.) / name. I remember that his colleagues calling him "Al.") |
|
256 / 12 |
(12/99) |
depression), / depression, |
|
/-1 |
(12/99) |
Blassenohl / Blessenohl |
|
257 / 2 |
(12/99) |
edged). / edged. However their methods were completely different and
they were able to extend my results in the certain cases of characteristic
p.) |
|
259 / 23 |
(12/99) |
undergrads. / undergrads. (Not that others were not supportive. Certainly
Orrin Frink, the Chair at Penn State during my two years there, maintained
and supported the highest academic standards.) |
| 259/
27 |
(2/01) |
chairman
/ chairman, Raph Hull (another of Dickson's students!), |
| 260
/ 13 |
(2/99) |
Gilman / Gillman |
|
/-17 |
(2/99) |
10/5 |
|
/-16 |
(2/99) |
mathematics. / mathematics. (In the Amer. Math. Monthly, 106 (1999)
p. 97, it is stated by Kenneth A. Ross in the "Distinguished Service
Award for Leonard Gillman" that "Len held a piano fellowship for five
years at the Juillard Graduate School before turning to mathematics.") |
|
262 /-5 |
(1/99) |
82 / 87 ("Orrin, or Frinky as I called him, was born in 1901, and
died in 1988" (letter from Aline Frink of December 20, 1998).) |
|
262 /-18 |
(12/99) |
Gaiété
Parisiennes / Gaieté Parisienne |
|
263 / 9 |
(12/99) |
Romantische / Romantisches |
|
/ 16 |
(12/99) |
rote
/ roten |
|
264 / 17 |
(12/99) |
Ruhren / Beruhigen |
|
264 /17 |
(1/99) |
Sovort / sofort |
| 265
/ 17 |
(12/99) |
Mög
Es / Mögen's |
|
265 / |
(1/99) |
insert
the following additional paragraphs before "Marston Morse":
Willy, The Heidelberg VW Salesman
Nevertheless,
by looking for and too often finding flaws in the national character
of Germans, I often wonder if we have not squandered a reservoir
of goodwill that many Germans felt for us and the Allies for relieving
them of the evident repressive evils of Nazism. Along with the bad,
many of the good suffered or perished under Hitler's repugnant regime.
I became good friends with the VW salesman whom I shall call Willy,
who in September '59 sold me a 1953 VW "Bug" for the then magnificent
sum
of $600, or about 2400 Deutsche Marks. (The Dollar was King back
then!) He told of his being "captured", i.e., surrendering to the
Allies along the Rhine, and being roughly interrogated by an American
who happened to be Jewish. Although Willy was fully cooperative
and eager to please, the interrogator, after finding a photograph
of his wife and family in Willy's wallet, tore it up in front of
him. When I expressed my deepest sympathy at this bit of cruelty,
he replied, "Ja, Herr Professor, but you must remember we Germans
tore not merely photographs but their people to pieces!" This admission
of collective guilt and the frightful imagery moved me to tears.
I invited him to my home in Neuenheim, and subsequently we exchanged
family visits throughout the academic year 1959-60.
Italiensche
Reise
In Spring recess, March l959, my first wife, Mickey, and my daughters,
Heidi and Cindy, two German babysitters, and I (together with our
luggage under the front hood), crammed into our tiny "beetle"for
a three week exploration of the South from Heidelberg to Freiberg,
Basel, Zurich, the Jungfrau, Lausanne, Geneva, the French Alps,
Lyon, Marseilles, Nice, Cannes, Rapallo, Pisa, Siena, Gaeta, Rome,
Pompeii, Herculaneum, Ravenna, Venice, the Dolomites, Brenner Pass,
Innsbruch, the Arlberg, Konstanz, Schaffhausen and back via Mannheim
to Heidelberg. (Not bad for a $600 car?) One of our babysitters,
tall and blond, attracted a great deal of admiration in macho Italy.
Once we had to return a radio given to her as a present, when the
ardent lover had his ardor doused by our curfew. Another time, when
we returned from viewing the Coliseum, we saw them surrounded, again
by machismo, while the children teetered on the curbside unattended
with speeding cars whizzing by. ("What does not destroy you makes
you stronger." -- Nietzche) |
| 265/-3 |
(1/99) |
society / Society |
|
265 /-1 |
|
|
|
266 / 17 |
(12/99) |
apochryphal / apocryphal |
| 268
/ 3 |
(12/99) |
comprised of / comprising |
|
269 /-10 |
(12/99) |
Gärding/
Gĺrding |
|
270 / 2 |
(12/99) |
in Amsterdam / at Harvard |
|
/ 4 |
(12/99) |
of / off |
|
274 / 12 |
(12/99) |
Adolph / Adolf 274n |
|
/-4 |
(1/99) |
later / later" |
| 274
/-4 |
(1/99) |
forgot]
/ forgot |
|
/-4 |
(1/99) |
nazi."
/ Nazi.] |
|
275 /-10 |
(12/99) |
Schoenfliess / Schoenflies |
|
277 /-9, -14 |
(12/99) |
Rudy / Rudi |
|
278 / 5 |
(12/99) |
en / ein |
|
278/11 |
(1/99) |
(x) / powers of x |
|
278 /-17 |
(12/99) |
Rockerfeller
/ Rockefeller |
|
278 /-2 |
(12/99) |
genuises
/ geniuses |
|
279 /-2 |
(12/99) |
apochryphal / apocryphal |
| 279
/-21 |
(12/99) |
"Hörgelt"
is Yiddish. The German word is Hörgeld |
| 280 |
(5/00) |
Wooster
/ Worcester |
|
281 /-6 |
(3/99) |
3.33 / 13.33 |
| 281
/-1 |
(7/99) |
To Footnote 19, add: According to Paul Halmos (in Amer. Math. Monthly,
1973, & reprinted in the World Treasury of Physics, Astronomy
and Mathematics, Timothy Ferris (ed.), Little Brown (Back Bay), Boston,
1991), Maniac is an acronym for Mathematical Analyzer, Numerical Integrator,
Automatic Calculator! However, the first general purpose electronic
computer was the Eniac, theoretically, based on Alan Turing's "machine",
or mathematical description of a machine, that in principle could
solve general mathematical equations. The Eniac (=electronic numerical
integrator and computer) was devised by P. Eckert and J.W. Mauchly
at the University of Pennsylvania in WWII and became operational in
1946, that is, after the war. Herman Goldstine (N.B.) is the sole
surviving member of the Eniac team. |
|
283 / 12 |
(1 /00) |
abroad / abroad. In "Duino Elegies," Rilke has this to say about leave
taking:
"Who turned us around like this so that we always,/do what we
may, retain the attitude of someone who's departing? Just as he /
on the last hill that shows him all his valley / for the last time,
will turn and stop and linger, / we live our lives for ever taking
leave. " |
|
283 /-7 |
(2/99) |
E.
Kolchin / Ritt |
|
283 .-12 |
(12/99) |
whiskey
/ whisky ("whiskey" is the American and the Irish sort) |
|
285 /-5 |
(12/99) |
Herman / Hermann |
|
287 / 5 |
(12/99) |
which / which every |
|
/ 11 |
(12/99) |
at (x) / at powers of x |
|
287 /-20 |
(12/99) |
17.40 ff / 7.40 ff |
|
287 /-17 |
(2/99) |
John's
/ Johns' |
|
288 /-11 |
(12/99) |
Hembda / Hemda |
|
289 / 1 |
(12/99) |
uncrossed / "uncrossed" (or noncrossed) |
|
289 / 2 |
(12/99) |
~2 / 2.4D |
|
289 / 3 |
(12/99) |
Shimson / Shimshon |
|
291 / -4 |
(2/99) |
A. Pedoe / D. Pedoe |
|
291 /-2 |
(12/99) |
New England / England |
|
292 /-17 |
(12/99) |
Chandigahr / Chandigarh |
|
294 / 20 |
(2/99) |
Mahabili Purim / Mahabalipuram (Seven Pagodas; also called Mamallipuram,
or city of Malla) |
|
295 / 2 |
(2/99) |
Seven Wonders / Wonders |
|
295 / 21 |
(12/99) |
Shackelton / Shackleton |
| 295
/ 25 |
(12/99) |
Penquin
/ Penguin |
|
295 |
(2/99) |
Add to Footnote 25: The Seven Wonders of the World were talked for
centuries B.C. (See, e.g., Leonard Cottrell, "Wonders of the World",
1959.) On the other hand, the Taj Mahal was completed in 1648. (See
Columbia Encyclopedia, which asserts that the jewels on the exterior
were "semiprecious stones".) |
|
296 / 14 |
(4/99) |
Fisher / Fischer |
|
297 / 12,15 |
(3/99) |
Nietsche/ Nietzsche |
|
298 / 21 |
(12/99) |
forebearer
/ forebear |
|
299 / 6 |
(12/99) |
Rheinhold / Reinhold |
|
299 / -12 |
(3/99) |
A.M.S. by Jane Scanlon / A.M.S. (Vol.11) by Jane Scanlon (Cronin) |
|
301/16 |
(1/99) |
225 / 250 |
|
301/-3 |
(1/99) |
Gröthendieck
/ Grothendieck |
|
302/ |
(1/99) |
Preceding "The Gorenstein Report and "Dream Time", insert:
Richard Brauer and the Postcard from Balestrand
While
at the University of Kentucky, I learned that Richard Brauer, one
of the many mathematicians displaced by WWII in the "Intellectual
Migration", had been a visiting professor there several years earlier.
But it was not until I attended an AMS meeting at Harvard in 1960
that I met him personally: I dropped by his office for a chat. (
I think he was the chairman, at least his office had 'chairman'
size!) After some time, I became aware of my imposition when he
apologized to me for taking up so much of my time. (Was my face
red!) I heard him lecture at the Institute for Defense Analyses
in Princeton in the summer of 1964, on the subject of Simple Groups.
Brauer was the humblest mathematician I ever met. He thanked you
after his IDA lecture as if you were doing him a favor to attend
his stimulating lecture! After the ICM in Stockholm in 1962, we
visited Norway: Oslo, Bergen, Flam, and Balestrand on the Sogne
Fjord. As I walked onto the ferry leaving Balestrand after viewing
the glacier there, I bumped into Brauer and his wife. When I got
back to the Institute for Advanced Study, I found a postcard from
Brauer posted in Balestrand! (I mentioned this bit of inspiration
in my Springer Algebra.) |
| 303
/ |
(2/99) |
Insert directly preceding "Ken Goodearl et al":
Danny's incredulity was later echoed by a
real estate broker I knew who belatedly found out that I owned a house
on the lake: "You have a house, on the lake? (Previously had she thought
I was a mathematical poor mouse? As a matter of fact, we do have the
cheapest house on the lake!) |
|
305 / 19 |
(12/99) |
Terra / Terre |
|
/ 23 |
(12/99) |
Johannesberg / Johannesburg |
| 306
/ 3 |
(12/99) |
Fall 1989. / Fall 1989. (Universitat Autónoma de Barcelona
(UAB) is the official name.) |
|
309 / -11 |
(2/99) |
Pere Menal, / Pere Menal (and many, many others!) |
|
312 /-10,-11 |
(4/99) |
Preceding "Kap", add: John Cozzens almost immediately applied
Kap's advice: He came to me one day and said, "Carl, I could get the
counterexample to Bass' question [see p. 287] if only I could find
a differential field over which I could solve these blankety- blank
equations." I knew then that John had his Ph.D. because I had read
Kap's monograph [57] in which I learned of Kolchin's universal differential
fields. |
|
312 /-7 |
(2/99) |
philosophy / engineering |
|
/-3 |
(1/99) |
assuming / following as director |
|
/-2 |
(1/99) |
of the / of S.S. Chern |
|
/-2,-1 |
(12/99) |
relinquished to David / relinquished to William Thurston followed
by David |
|
313 / 1 |
(2/99) |
Add: A stimulating introduction to Kap's mathematical life and thought
appears in More Mathematical People, (See Bibliography, Albers et
al., eds.) |
|
317 /-23 |
(12/99) |
Nauseé
/ La Nausée ; L'Etre et le Neant / L'Ętre et le Néant |
|
317 /-4 |
(12/99) |
Mediterannean / Mediterranean |
|
320 /-3 |
(4/99) |
Rights / Rights. (See "Berkeley in the Sixties", 1990, documentary
by film maker Mark Kitchell.) |
|
321 /6 |
(5/00) |
Republican
/ Democratic |
|
321 / 19 |
(2/99) |
in
1980 / in 1966 |
|
321 |
(2/99) |
After the Steve Smale paragraph, add:
Steve Smale appears in More Mathematical People (See Bibliography,
Albers et al., eds.), and Steve Batterson [00] has written a full
scale biography of him. |
|
322 / -10, -11 |
(2/99) |
Insert:
"Beauty is the first test: there is no place in the world for ugly
mathematics" -G.H. Hardy in "Mathematician's Apology (1940)
"The union of the mathematician with the poet, fervor with measure,
passion with correctness, this surely is ideal." -William James, 1879,
quoted in Bartlett's |
| 322
/ -4,-5 |
(2/99) |
Insert: This echoes Einstein: "Science will stagnate if it is made
to serve practical goals." (Quoted in Calaprice's "The Quotable Einstein,
p. 181.) |
|
323 / -22,-12 |
(2/99) |
Kluvver / Kluwer Bertram / Bertrand |
|
324 /-1 |
(11/99) |
This echoed Paul Valéry's dictum with "Work of art' replaced by "poem."
See p. 106 in "Quotationary," L.R. Frank ed., Random
House, New York, 1998. Unfortunately I cannot justify my attributing
this to Picasso; however many artists must have said the same
thing. |
|
324 |
(6/00) |
Add:
"Sex if the sublimation of the desire to do mathematics."
Berkeley Grafitto (1965-6)
"It's
better to have both a spouse and a lover because one will think
you are with the other when you are having mathematics."
Mathematics Alibi, contributed by a colleague who has
neither. |
|
S - 1 / 11 |
(3/99) |
299, 315 / 275 |
|
S - 1 / 16 |
(2/99) |
295
/ 296; Also under Cherlin, Donnelly, Menal, Miller, O'Neill:
xxix / xxx |
| S
- 1 /-6 |
(12/99) |
Blassenohl
/ Blessenohl |
|
S - 1 /-3 |
(12/99) |
Bollabas
/ Bollobás |
|
S - 2 |
(1/99) |
under "Faith", add Heidi 268 |
| |
(2/99) |
under
"Dukas", add 314n |
| |
(12/99) |
Erdös
/ Erd s |
|
S - 3 / 13 |
(2/99) |
To "Hitler", add 304 and change Adolph to Adolf |
| |
(2/99) |
Gröthendieck
/ Grothendieck |
| |
(2/99) |
Add:
Grove, Vernon 262 |
|
S-3 / 2 |
(12/99) |
Change
the diacritical mark to a single dot Gärding/Gĺrding |
|
S-3 / -12 |
|
Jaccobson
/ Jacobsson |
| S-3
/-9 |
(12/99) |
Florrie
/ Florie |
|
S-3 /-5 |
(12/99) |
Under "Hardy": 265 / 266 |
|
/-3,-4 |
|
Add: Hass, see Whitney |
|
S-4 |
(2/99) |
To "Maniac", replace 280 by 281 and add 316 |
| |
(12/99) |
Kennen
/ Kennan |
|
S - 4 / |
(2/99) |
Add:
Kingsley, Ben 296n |
| |
(2/99) |
Under
"Kelley": 261 / 262 |
| |
(5/99) |
Add: Lawrence, John 292 |
|
S
- 4 / -12 |
(2/99) |
Karl / Carl |
| |
(12/99) |
MacLane
/ Mac Lane |
|
/ -3 |
(2/99) |
To "Morse", add 310 |
|
S-5 |
(12/99) |
Under
"Rentschler" : Rudolph / Rudolf |
|
S-5 |
(2/99) |
To "Ritt", add 283 |
|
/ -13 |
|
Page,
S. 287 / Page, S. 288 |
|
/ -13 |
(2/99) |
Bertram
/ Bertrand |
|
S
- 6 |
(3/99) |
To "Selberg, Atle", add 290 |
| |
(12/99) |
Shackelton / Shackleton |
| |
(9/99) |
Add:
Shanks, Nelson 260 |
|
S-6 |
(2/99) |
Add
Silverman, Ed 262 |
|
/ -15, -16 |
|
Insert:
Stevenson, Robert Louis 295 |
|
S-6 |
(2/99) |
To "Sullivan," add: 263, 278 |
|
S-6 |
(1/99) |
To "Utumi", add 312; Add: Visonhaler, Charles 305 |
| S-7
/ 15 |
|
Wood,
Japheth 291 / Wood, Japheth 291, 322 |
|
/ 3 |
(1/99) |
To "Whitney" add: 270 |
| |
(12/99) |
Under
"Witt": Ernest / Ernst |
|
S-7 /-12 |
(1/99) |
Yang, Ning Chang / Yang, Chen Ning |
|
325 |
(2/99) |
Add: [90] D. Albers, G.L.Alexanderson and C. Reid, More Mathematical
People, Harcourt, Brace, and Janovitch, Boston, San Diego, and
New York, 1990 |
| |
(2/99) |
Add: [90] G.L. Alexanderson, see Albers |
| 326 /
7 |
(12/99) |
Kertesz
[71] / Kertesz [72] |
|
326 |
(1/99) |
Add: [98] P.N. Ánh, D. Herbera, and C. Menini, AB-5* for module
and ring extensions, pp. 59-68 in [98] Dikranjan and Salce. |
|
326/
-8 |
|
[95]
/ [97a] |
|
326 / -6 |
(2/99) |
[93]
/ [97b] |
| |
(2/99) |
(1988??) / 197(1997) 409-423 |
|
/ -3 |
|
[97] / [98] |
|
/ -2 |
|
Math.
(1998) / Math. 105(1998) 105-137 |
|
327 / 21 |
(12/99) |
Hambrug
/ Hamburg |
|
327 / 22 |
|
[50] / [51] |
|
327 /-12 |
(12/99) |
Ashsbacher's review
of Gorenstein, et.al. appeared in the Notices of the A.M.S., Providence, 1997 |
|
328 / 2,3 |
(12/99) |
Add: [96] G. Azumaya, A characterization of coherent rings in
terms of finite matrix functors, pp. 1-3 in the Second Japan-Asia
Symp. on Ring Theory and the Twentieth Symp. on Ring Theory at
Okayama U, 1995, Okayama, U. Press, 1996 |
|
328 / |
(12/99) |
Add: [36] R. Baer, The subgroup of elements of finite order of
an abelian group, Ann. Math. 37 (1936), 766-85 |
|
328
/ |
(12/99) |
Under "Ballet": lineares' / linéares |
|
328 / |
(2/99) |
Add: [99] I.G. Bashmakova and G.S. Smirnova, The Birth of Literal
Algebra, Amer. Math. Monthly 106 (1999) 57-66; transl. from the
Russian by A. Shenitzer. (Translator's note: this is the third chapter
by the authors devoted to the evolution of algebra. See loc.cit.,
p.57) |
| 328
/-4 |
|
[60] ______,
/ [60] H. Bass, |
|
329 |
(2/00) |
Add: [00] S. Batterson, Stephen Smale, the Mathematician Who
Broke the Dimension Barrier, Amer. Math. Soc., Providence, 2000. |
|
329 / 19 |
(12/99) |
[69] / [71] ...1977 / 1991 |
|
330 |
(9/99) |
Add: [74] J.W. Brewer and W. Heinzer, Associated primes of prin-cipal
ideals, Duke Nath. J. 41 (1974) 1-7 |
| 331
/ 12 |
(2/99) |
Delete "[80]______, see Atkins." |
|
332 |
(4/99) |
Add: P-J Cahen, M. Fontana, E. Houston, and S-E Kabbaj, (eds.),
Commutative Ring Theory, Proc. of the II Intern. Conf. (Fes,
l995), Lect. Notes in Pure & Appl. Math., vol 185, M. Dekker
Inc., Basel and New York, 1997. 332 |
| |
(1/99) |
Add: [96] A. Calaprice, The Quotable Einstein. Princeton U. Press,
Princeton, N.J., 1996 |
|
332 / 11 |
(2/99) |
polynomail /
polynomial |
|
/ 24 |
(2/99) |
nnote
/ note |
|
332 |
(4/00) |
Add: [78b] V.P. Camillo, On a conjecture of Herstein, J. Algebra
50 (1978) 274-5. [78] V.P. Camillo and J. Zelmanowitz, On the
dimension of a sum of modules, Comm. Algebra 6 (1978) 345-52 |
|
332 |
(2/00) |
Add: [72] A.B. Carson, Coherence of polynomial rings over semi- simple
algebraic algebras, Proc. A.M.S., 34(1972) 20-24. |
| |
(4/00) |
[76]
A. B. Carson. Representations of regular rings finite index,
J. Algebra 39 (1976) 512-26. [78] A.B. Carson, Coherent polynomial
rings over regular rings of finite index, Pac. J. Math. 74 (1978)
327-32. |
|
333 / 6 |
(3/00) |
[95]
____, / [94] ____, |
|
333 / 11,12 |
(2/00) |
[95] /
[98] ... Math. Soc. (1995) / Math. Soc. 126 (1998) 2541 - 48 |
|
333 |
(3/99) |
Add: [84]
S.U. Chase, Two remarks on central simple algebras, Comm. Algebra
212(1984) 2279-89 |
|
333 |
(4/00) |
Add: A.W. Chatters, A nonsingular Noetherian ring need not have
a classical quotient ring, J. London Math. Soc. 10 (1975) 66-68. |
| 333 |
(1/99) |
[80] A.W. Chatters / [80] A.W.Chatters and C.R. Hajarnavis |
|
333 / 14 |
(2/99) |
Ammsterdam / Amsterdam |
|
334 / |
(1/00) |
Add:
[00] S. Batterson, Stephen Smale: The Mathematician Who Broke
the Dimension Barrier, Amer. Math. Soc., Providence, 2000 |
|
335/-12 |
(3/99) |
mene / meme |
|
336 /1 |
(1/99) |
nd
/ und |
|
/ 19 |
|
Beneath
[93] W. Dicks insert: [94] _____, see Menal. |
|
/ -11 |
(4/00) |
Topology,
Lecture / Topology, Padova Conf. 1997 in honor of Adalberto Orsatti's
Sixtieth Birthday Lecture, Lecture Notes in Pure and Applied Math,
Vol. 201, Marcel Dekker |
|
/ -10 |
|
applied Math., Marcel / Applied Math., vol.201, Marcel |
|
/-1 |
(4/00) |
M.
Domokos, Goldie's Theorems for involution rings, Comm. Algebra
22 (1994) 371-80 |
|
337/ |
(1/99) |
Add: [79] H. Dukas and B. Hoffman, Albert Einstein: The Human
Side, Princeton U. Press, 1979 |
|
338 |
(3/99) |
Add: [98] A. Facchini, Module Theory, Progress in Math., vol.
167, Birkhauser, Basel, Boston and Berlin, 1998 |
|
341 / 1 |
(2/99) |
37-43.
/ 37-43. (See Example 9.17 in the text.) |
|
/ 18 |
(11/99) |
Delete "II" |
|
/-10 |
(3/00) |
ascending
chain conditions / acc |
|
/ -9 |
|
Math., /
Math., vol.201, |
|
341 /-7 |
(2/99) |
[98B]
/ [99] |
|
/-6 |
(2/99) |
Algebra (1998) / Algebra 27 (1999) 1807-10 |
|
341 |
(2/00) |
Add: [00a] C. Faith, Note on residually finite rings, Comm. Alg.
28 (2000) 4223-26.
[00b] ______, Associated primes in commutative polynomial rings,
Comm. Alg. 28 (2000) 3983-86.
[00c] _____, Coherence and annihilators in matrix and polynomial
rings, preprint (2000) |
|
342 / 1 |
|
[1960]
/ [1965] |
|
343 / 7 |
(11/99) |
[74] / [75] |
|
343 |
(4/99) |
Add: [97] M. Fontana, see Cahen [97] |
| |
(2/99) |
Add: [90] E. Formanek, The Nagata-Higman Theorem, J. Appl. Math.
21 (1990) 185-192. Italicize the entire title of [14] Fraenkel. |
|
343/-14 |
(1/99) |
Math.
/ Math. Society |
| |
(2/99) |
Add: L. Fuchs and S. Shelah, On Kaplansky's problem on valuation rings, Proc.
Amer. Math. Soc. 105 (1989) 25-30 |
|
344 /-15,-13 |
(2/99) |
L. Gilman / L. Gillman / 1,2 (2 /99) preprint, 1998 / Colloq.
Math. 1998 or 9. |
|
344 |
|
Add: [87] R. Gentle, Comment on the Nullstellensatz for regular
rings, Canad. Bull. Math. 30 (1987),
124-128. |
| |
(2/99) |
Add: [65] J.W.L. Glaiser (ed.), see H.J.S.Smith |
|
345 /-3 |
(4/00) |
selfinjective
/ self-injective |
|
346 / 16 |
(4/00) |
Non-commutative Rings / Non-Commutative Noetherian Rings |
|
346/-7 |
(1/99) |
envveloppe
/ enveloppe |
|
347 / |
(2/99) |
Gröthendieck /
Grothendieck |
|
347/ |
(1/99) |
L.Gruson / L. Gruson and M. Raynaud |
|
347 / |
(2/99) |
Add: [73] M. Gruson and C.U. Jensen, Modules Algébriquement compacts
et foncteurs limî, C.R. Acad. Sci. Paris Se\'r. A-B 276 (1973)
127-131 |
|
348 |
(2/00) |
Add: [67] M. Harris, Some results on coherent rings, Glasgow
Math J.8 (1967) 123-6 |
|
348 |
(4/99) |
to Heinzer, add: see Brewer [74] |
|
/18 |
(6/00) |
[74]
/ [75] |
|
/21 |
(6/00) |
[96]
/ [69,00] |
|
/-8 |
(6/00) |
Tokyo, 1996 / Tokyo / 1996, 2000 |
|
349 /-9 |
|
Add:
[96] D. Herbera and A. Shamsuddin, On self-injective perfect
rings, Canad. Math. Bull. Vol.39 (1996), 55-58 |
|
349 |
(6/00) |
Add:
[96] M. Hazewinkel (ed.), Handbook of Algebra, North / [96,00] M.
Hazewinkel (ed.), Handbook of Algebra, vols. 1 and 2, North |
|
350 |
(5/99) |
[98] Y. Hirano, On rings over which every module has a maximal sub-module,
Comm. Algebra 26 (1998) 3335-45 |
| |
(5/99) |
[91] Y. Hirano and J.K. Park, Rings for which the converse of
Schur's Lemma holds, Math. J. Okayama.U. 33(1991) 121-131
[95] Y.Hirano, C.Y. Hung, and J.Y. Kim, On strongly bounded
and duo rings, Comm. Algebra 23 (1995) 2199-2214. |
|
350
/ 10 |
(12/99) |
[56] ------, / [56] Graham Higman, |
|
350 |
(4/99) |
Add:
[97] E. Houston, see Cahen et.al., (eds.) |
|
354 |
(4/00) |
Add: [69] R.E. Johnson, Extended Mal'cev domains, Proc. A.M.S.
21 (1969) 211-213. |
|
355 |
(4/99) |
Add: S-E. Kabbaj, see Cahen, et.al. (eds.) |
|
356 / 1 |
|
in
Collected / in Selected |
| 356 |
(1/01) |
Add:
F. Kasch, Moduln und Ringe, B.G. Tueber, 1977 |
| |
(12/00) |
Add:
Karamzadeh, On a question of Matlis, Comm. Alg. 25 (1997) 2717-2726. |
|
356/ |
(6/99) |
Add: O.A.S. Karamzadeh, and A.A. Koochakpoor, On Aleph-null
self- injectivity of strongly regular rings, Comm. Algebra 27(1999)
1501- 1513 |
| |
(2/99) |
Add:
Y.Kawada, On similarities and isomorphisms of ideals in a
ring, J. Math. Soc. Japan 9 (1957) 37-4-80 |
| |
(12/99) |
Add: I. Kleiner, "Field Theory": from equations to axioms, Part
II, Ibid. 106 (1999) 859-63. |
| |
(6/00) |
Add: [00] N.Ya. Komarnitskii, The Cozzens-Faith problem
on "Countable" ultrapowers of the Koifman-Cozzens
domain, Math.Stud. 7 (1997), 3-26, 11 |
|
/-5 |
(6/00) |
[96]. K. Kharchenko / [96] V.K. Kharchenko |
| |
(6/00) |
Add: [00] V.K. Kharchenko, Fixed rings and non-commutative invariant
theory, in Hazewinkel, [00], pp. 359-98. |
|
357 |
|
Add: [91] Y. Kitamura, Inheritance of FPF rings, Comm. Algebra 19(1991),157-165. |
| |
(12/00) |
Add:
[95] J.Y. Kim, see Hirano |
|
358/-1 |
(1/99) |
L'Enseg. / L'Enseig. |
|
358 |
(2/99) |
Add: [99a] T.Y. Lam, Bass's work in Ring Theory and Projective
Modules, pp. 83-124 in Lam and Magid [99] . |
| |
|
[99b]
_______, Letter to the author of January 20, 1999. |
| |
(2/99) |
Add:
[99] T.Y.Lam and A.R Magid (eds.), Algebra, K-Theory, Groups
and Education, On the Occasion of Hyman Bass's 65th Birthday,
Contemporary Math., Amer. Math. Soc., Providence, 1999. |
|
359 |
(3/00) |
Add: [76] J. Lawrence, When is the tensor product of local algebras
local? II, Proc. A.M.S. 58 (1976) 22-24 |
|
360 |
(4/99) |
Add:
[99] C. Lomp, On semilocal modules and rings, Comm. Algebra
27 (1991) 1921-35 |
| 361 |
(12/99) |
Maclane / Mac Lane |
|
361 / |
(3/99) |
Add:
[99] A. Magid, see Lam |
| |
(2/99) |
Add:
[60] J.M. Maranda, On pure subgroups of abelian groups, Arch.
Math. 11(1960) 1-13 |
|
361-73 /-12 |
(1/99) |
[75]
/ [75b] |
|
362 / -5 |
(2/99) |
[86]
/ [88] |
| 362
/-6 |
(2/99) |
[82] / [82b] |
|
363 |
|
After
P. Menal & J. Moncasi, |
|
363 |
|
Add:
[82] P. Menal and J. Moncasi, On regular rings of stable range
2, J. Pure and Appl. Algebra 24(1982), 25-40 |
|
364 / 1 |
|
move
the reference "[90] S.H. Mohamed and B. J. Mueller" to p.
363 to directly precede the reference "[89] A. Mohammed and F.L.
Sandomierski" |
|
365/-12,-13 |
(1/99) |
Add:
[75a] W.K. Nicholson, I-rings, Trans. A.M.S. 207 (1975) |
|
366 |
|
Add: [99] B. Olberding, Almost maximal Prüfer domains, Comm.
Algebra 27 (1999) 4433-58. |
| 365 |
|
Add:
W.K. Nicholson and M.F. Yousif, Mininjective rings, J. Algebra
187 (1997) 548-78. |
| 368 |
(10/00) |
Add:
[99] B.L. Osofsky, Nice polynomials for introductory Galois Theory,
Math Magazine, 72 (1999) 218-222. |
| 368 |
(6/00) |
Add:
[99] E. Osmanagic, An approximation theorem for Krull rings with
zero divisors, Comm. Algebra 27 (1999) 3647-57 |
|
370 / 1 |
|
Move
the reference to precede: "[23] H. Prüfer" |
|
370/ |
(6/99) |
Add:
R.Raphael, Injective rings, Comm. Algebra 1 (1974), 403-14 |
| 370/-12 |
(1/99) |
Make
Gruson the first author, and move the reference to p. 347. |
|
/-12 |
(1/99) |
Add:
M. Raynaud, see Gruson |
|
/-6,-7 |
(2/99) |
Add:
[90] C. Reid, see Albers 372 |
| |
(3/00) |
Add:
[90] M. Roitman, On polynomial extensions over countable fields,
J. Pure and Appl. Algebra 64 (1990) 315-28 |
|
371 |
(6/00) |
Add.
[99] P. Ribenboim, The Theory of Classical Valuations, Springer
Monographs in Math, Springer Verlag, NY, 1999 |
|
372 |
(11/99) |
Add:
[71] E.A. Rutter, PF-Modules, Tohoku Math.J. 23(1971) 201-206 |
|
372 /-12 |
(3/00) |
Comm.
Alg. 3 / Pacific J. Math.51 |
|
373 /-3 |
(3/99) |
Delete:
[67] ___, see Cohn |
|
373 /-14 |
(2/99) |
See
Brewer / See Eakin |
|
373 |
(4/00) |
Add: [33] F.K. Schmidt, Mehrfachperfekte Körper, Math. Ann. 108
(1933) 189-202 |
|
374 / 22 |
|
[95]____, see Herbera / [95,96]______, see Herbera |
| 374 |
(2/99) |
Add:
[99] A. Shenitzer, see Bashmakova and Smirnova |
|
374 |
(2/99) |
Add: [71c] R.C. Shock, Injectivity, annihilators, and orders, J. Algebra
19 (1971) 96-103 |
|
374 |
(2/99) |
Add:
[72b] R.C. Shock, The ring of endomorphisms of a finite dimensional
module, Israel J. Math 11 (1972) 309-14 |
|
374 |
(2/99) |
Add:
[74] T.S. Shores, Loewy series of modules, J. angew. Math
265 (1974) 183-200 |
|
376 |
(2/99) |
Add: H.J.S. Smith, Collected Math. Papers (J.W.L. Glaiser, ed.), 2
Parts, AMS
Chelsea Pub., Providence, 1965. |
| |
(2/99) |
Add: [81] M.K. Smith, see Brewer (4/99) |
| |
(4/99) |
Add: Y.K. Song, Maximal commutative subalgebras of matrix rings, Comm.
Algebra
27 (1999) 1649-63 |
| |
(2/00) |
Add: [68A] J. Soublin, Anneaux cohérents, C.R.Acad.Sci. Paris
Sér. A-B 267 (1968) A 183-6. [68b]_____, Un anneau cohérent don't
l'anneaux des polynomes n'est pas cohérent, Ibid. A 241-3. |
|
376 |
(3/00) |
Add: [75] M.E. Sweedler, When is the tensor product of local
algebras local? I, Proc.A.M.S. 48 (1975) 8-10. |
|
377 / 16 |
(1/99) |
Epimorphisinen / Epimorphismen |
|
378 |
|
Add: [37] O. Taussky, see Jacobson |
| 378
/-11 |
(12/99) |
Trilfaj
/ Trlifaj |
|
379 |
(4/00) |
Add:
[94] A. del Valle, Goldie dimension of a sum of modules, Comm.
Algebra 22 (1994) 1257-1269. |
| 380 |
(12/99) |
Vicnair
/ Vicknair |
|
381 |
(2/00) |
Add:
J.J. Watkins, see Brewer |
|
382 / 2 |
(1/99) |
Lifting / Splitting |
|
383 |
|
Before "[73] T. Wurfel" add: [97] T. Wu and Y.Xu, On the stable range
condition of exchange rings, Comm. Alg. 25 (1997) 2355-63 |
|
383 / 17 |
(3/00) |
pp. 277-900 / pp. 277-300 |
| 383 |
(4/00) |
Add:
[97] W. Xue, Quasi-Hamsher modules and quasi-max rings, Math. J.Okayama
39 (1997) 71-79 |
|
383 |
|
Add:
[98 ] W.Xue, Rings related to quasi-Frobenius rings, Algebra
Colloq. 5 (1998), 471-480. |
| 383
/ 20,21 |
(4/00) |
[98]
/ [00] ... Fujian...Iowa City, 1998 / Comm. Algebra 28 (2000)
2633-2638 |
|
385/16 |
(1/99) |
H.Ziegler
/ M. Ziegler |
|
387 |
|
Birkhoff
33,240 / Birkhoff 33,241 |
| |
(4/00) |
To
"Camillo" add: 62 |
| |
(2/00) |
Add:
Carson 95 |
| |
(12/99) |
Capson
/ Copson |
| |
(3/99) |
To
"Cailleau", add 63 |
| |
(2/99) |
To "Camillo", add 131 |
|
/ -3 |
|
To "Cohn", add 129-30, 174 |
|
388 |
(4/00) |
Add
Domokus 62 |
| |
(9/00) |
To
"Fontana", add: 128 |
| |
(4/00) |
Evans
158 / Evans 157 |
| |
(3/99) |
Fuchs
6 / Fuchs 6n Add: Faith-Menal 65 |
| |
(2/99) |
Gabriel 90 / Gabriel 91 |
| |
(4/00) |
To Gilmer, add: P Add: Goldie 214 |
| |
(1/99) |
Add: Gómez Pardo 90n |
| |
(2/99) |
Goodearl 158 / Goodearl 157 |
| |
(2/99) |
Add: Goodearl - Warfield 211 |
| |
(2/99) |
Add: Guralnick 131 |
| |
(2/99) |
To
Hajarnavis, add 197 |
|
389 |
(11/99) |
Under
"Dung": 189 / 188 |
| |
(11/99) |
Under
"Huynh": 189 / 188 To "Harada" add 218 |
| |
(3/99) |
In "Hinohara": 109 / 108 |
| |
(2/99) |
To "Huckaba", add 134 |
| |
(2/99) |
Add: Saroj Jain 201; and S.K. Jain 132 |
| |
(2/99) |
To
"Kaplansky" add 6, replace 122 by 121, and add 157. |
| |
(2/99) |
Add Kawada 132
To "Kitamura" add 104 |
| |
(2/99) |
Klein 132 / Klein 131 |
| |
(12/99) |
To
Kleiner add: 15.4Bf. |
| |
(2/99) |
To "Koifman", add 69 |
| |
(6/00) |
Add: Komarnitskii 66 |
| 390 |
(4/99) |
To "Lam", add 157 |
| |
(2/99) |
Add: (López)-Permouth 132 |
| |
(2/99) |
To
"Matlis', add 64
To "Mueller" add 152 |
|
391 |
(2/99) |
Add: Norton 197
To "O'Neill" add 7-8
To "Osofsky" add 55
To "Perlis" add 252 |
| |
(2/99) |
Add:
Renault 69 |
| |
(2/99) |
To
"Rizvi", add 132 |
| |
(2/99) |
To
"Roitman", add 131 |
| |
(3/99) |
To "Salce", add 6n |
| |
(4/00) |
To "F.K. Schmidt," add 111 |
| |
(6/00) |
Add: Osmanagic 168 |
| |
(6/00) |
To "Ribenboim" add 111 |
|
392 |
(2/99) |
To "Shock" add 84 |
| |
(2/99) |
To
"Small" add 84, and replace 237 by 238 |
| |
(11/99) |
Under
"Wisbauer" 185 / 188 |
| |
(4/00) |
Add:
Valle 62 |
| |
(2/99) |
Add: Vitulli 167 |
| |
(2/99) |
To
"Warfield" add: see Goodearl |
| |
(4/00) |
Under "Wisbauer": 189 / 188 |
|
393 |
(2/99) |
To "Yousif", add 132 |
| |
(3/99) |
To "Zanardo", add 6n |
| |
(4/00) |
To "Zelmanowitz" add 62 |
| |
(1/99) |
Add:
Zel'manov 74 |
|
395 |
(12/99) |
add:
acc[P], 16.9Ds |
|
396 |
(12/99) |
Add: AMVR, see almost maximal valuation ring |
|
396 |
|
Above
"Arithmetic ring" add:  (E,R),
see 3.7A
Above "Auslander", add: Ass R, Ass*R , 6.39s, 16.11-12, 16.29A |
| |
(2/99) |
To "Artin, M.", add 15.8 |
|
397 |
(12/99) |
To "balanced" add: strongly _____, 5.54 |
| |
(9/00) |
To "Brandal", add: 6.19A |
| |
(8/99) |
To Bezout domain (ring) add: p.108ff. |
| |
(9/99) |
To
Brewer, add: - Heinzer, 6.39 |
| 398 |
|
After
"characteristic equation", add: characteristic of a ring, 1.2s |
| |
(8/99) |
To
"Camillo - Fuller", add 3.33 F,H |
| |
(4/00) |
Add. Carson --- Theorem 14.21 |
|
399 |
(2/00) |
To
"coherent ring" add 6.6 ff., 6.13, 14.20 |
| |
(2/99) |
To
"Cohn theorem", add 6.25. |
|
400 |
(2/99) |
To
"Cozzens-Faith theorem" add 7.7A, 7.12, and change 2.l6F to
2.6G. |
| |
(12/99) |
Under "crossed": 2.5Bff / 2.5Dff. |
| |
(11/99) |
Under CS module: 12.4s / 12.4Cs |
|
401 / 7 |
(2/99) |
inverse of, 13.1a / inverse of __, 13.1s |
|
401 |
(3/99) |
Under
"division" add: transcendental - ring, 2.6Gs |
| |
(2/99) |
Under "Eilenberg", add ___Rosenberg-Zelinsky theorem 14.9 402 |
| |
(11/99) |
Add:
FA, see finitely annihilated |
| |
(12/99) |
Add: f.a.e.c. = the following are equivalent conditions |
| |
(2/99) |
Under "Faith", make the same changes as above for Cozzens-Faith. |
| |
(2/99) |
Add:
FBN, see bounded |
|
402 / -11 |
|
Modulle / module |
|
403 |
|
(2/99) Under "field", add: transcendental _____,1.28f |
| |
(2/99) |
Under
"finite", add: module___, 8.G (p. 154) |
| |
(12/99) |
Add: fir, see free ideal ring semi ______, 6.31s Under "free"
add: _____ ideal ring (= fir) 6.31 |
| |
(11/99) |
Add: finitely annihilated (=FA) p. 202 |
| |
(12/99) |
To
"Fitting" add: definition, theorem, 2.29As |
| |
(10/99) |
Add: FSI, see fractionally.
Add: FTAG, P,1.6,1.15A |
|
404 |
|
Add: Fundamental Theorem of Abelian Groups, see FTAG. |
| |
(2/99) |
Gabriel
4.Cf / Gabriel 4.D(1) |
|
405 |
|
To "Gilmer", add: Abhyankar-Heinzer- - theorem, 10.5 |
| 405 |
(2/99) |
Add:
Göbel, see Dugas |
| |
(2/99) |
To
"Goldie" add: dual- dimension, 8.C,13.15C
To "Goldie theorem" add 6.29 |
| 405 |
|
To "Griffith - Eisenbud theorem" delete 3.5Af and add 5.3A,B
To Griffith" add: theorem 3.5Af
In "Harada" delete: 13.9D
Add: Herbera-Faith Theorem 13.10 |
| |
(12/99) |
Herbera-Pillay
Theorems 3.6B-G |
|
406 / 1 |
|
12.40
/ 12.4D |
|
406 / |
(12/99) |
Add: IBN, see invariant basis number |
|
406 / 3 |
(2/99) |
theorem,
1.26 / theorem, 1.26-7 |
|
/ -19 |
|
To "(=prindec)" add: p. 86 |
|
/ -14 |
|
nilpotent-
/ nilpotent-, 3.80 |
|
407 |
(4/00) |
Under
"involution" add 3.13s |
|
407 |
(2/99) |
Add:
Ishii, see Harada
To "Jategoankar" add: see Formanek |
|
408 |
(4/00) |
To "Johnson" add: -- Mal'cev domain, 6.27f
Under Jøndrup: 7.,
7.3s / 7.3A |
|
408 |
(12/99) |
To "Kaplansky - Levitzki": 15.4 - should be 15.11 |
| |
(2/99) |
To
"Kaplansky theorem," add: 6.3A, 14.34 |
| |
(2/99) |
Add:
Kertész theorem, 1.27 To "Kitamura theorem" add: 4.32 |
| |
(2/99) |
To
"Klein", add 6.35 |
| |
(2/99) |
Add:
Koifman theorem, 3.21s, 3.32ff |
|
409 |
(12/99) |
Add:
Lasker ring, 2.29f |
|
409 |
(12/99) |
To "Levitzki" add: ___ Kaplansky Theorem, 15.11 |
|
409 |
(2/99) |
To
"Lie algebra", add: 14.47 |
| |
(2/99) |
To
"linear", add: full___ ring, 4.6 |
|
410/-3 |
(3/00) |
3.33 as / 3.33 As, 3.33 F-H |
|
410 |
(2/99) |
Add: López-Permouth, see Al Huzali; also Huynh; also Jain |
| |
(12/99) |
MacLane
/ Mac Lane To "Megibben theorem": 3.7C instead of 3.7 |
|
411 |
(2/99) |
Under "module", add: finite, 8.G |
| |
(4/00) |
Add:
Mori domain 9.4s |
|
412 / 2 |
(2/99) |
To locally___, 7.23, add: 9.1s (Note locally Noetherian is used
in two different senses: in 9.1s it means that every local ring is
Noetherian.) |
|
412 /-3,-4 |
(11/99) |
Under
"nilpotent" add: transfinitely -, 3.34A, 13.7, 13.7s |
| |
(9/00) |
Add:
Olberding theorem, 6.19A |
|
413 |
|
To "Ornstein" add 2.19B |
| |
(2/99) |
Add:
PI = (polynomial identity) central -, 15.6, Chap. 15 Notes
see polynomial |
| |
(1/00) |
Under Pillay Theorem: 3.6a-E / 3.6A-E |
| |
(12/99) |
Add: PP(= pp) ring, 7.3As
To "prime associated" add: 3.16Cs, 6.39s
To "prime minimal ideal" add: 2.37G, 14.34s, 16.25f (Remark) |
|
414 / 3,4 |
(2/99) |
Add: Spectrum = Spec, 14.1s |
| |
(2/99) |
Add: Procesi theorem, 4.14s, 15.8 |
| |
(2/99) |
Add:
Q(R) = classical quotient ring of commutative R. |
|
415 |
(9/99) |
Under "regular' add: semi-ideal, 9.28s |
| 415
/ -7 |
|
"rank
of a ring" : 3.56 instead of 3.56s |
|
415 |
|
Reis-Viswanathan: delete: 16.As, |
| |
(2/99) |
Add: Renault theorem, 3.32 ff (p.69) |
| |
(9/99) |
Under "Resco," add: theorem; 2.6H(2) |
| |
(2/99) |
Add: RD (= relatively divisible), see "pure" |
| |
(3/99) |
Under
"refinement", add: isomorphic --, 8.3f (p. 155) |
|
416 / 8 |
(2/99) |
Camillo
-, 3.33',5.20 / Camillo -, 3.4As (p.54) |
|
/9 |
(2/99) |
Osofsky -, 3.33' / Osofsky -, 3.4As (p.54) |
|
/ 10 |
(3/99) |
chain --, 3.14As / chain --, 3.14As, 5.52s (p. 119) |
|
416 |
|
Add: duo-,5.45 (2/99) Add: FBN ____, 3.36Es, 15.17s |
| |
(2/99) |
Add:
Lasker -, 2.29s |
| |
(2/99) |
Add: linear ___,4.6 |
| |
(11/99) |
Add:
star ring 6.60s |
| |
(1/99) |
Under "Roggenkamp" add: "-- Scott theorem 11.11f." |
| |
(1/99) |
Under "Roitman", delete "-- Scott theorem 11.10 ss" |
| |
(2/99) |
Under
"Rosenberg", add: see Eilenberg |
| 417 |
(10/99) |
Add:
star ring, p. 137  -ring
6.60s |
|
418 / 3 |
|
3.33as / 3.33As |
|
419/-7 |
8/99) |
see Amitsur, Goldie/ see Amitsur, Goldie, Herstein |
|
419 |
(2/00) |
Add: Soublin - theorem and example 14.20s |
|
419 |
(2/99) |
Add: Spec (= Spectrum) Spec R, the set of all prime ideals of
R Prime ---, 14.1s mspecR, the set of all maximal ideals of
R (= max spec R) |
|
420 |
(2/99) |
Add: Tominaga theorem, 1.27 |
|
421 / -9 |
|
Under
"Warfield theorem": 6.5a / 6.5A |
|
/-16 |
(3/99) |
__ - ring, 3.15as, 5.1A'f / __ - ring, 3.15As, 5.A'f, 5.52s |
|
421 |
(3/99) |
Under "valuation", add: generalized discrete ___, 9.35f |
| |
(2/99) |
Add:
VD = valuation domain
VR = valuation ring |
|
422 |
|
Zacharias, [88] / Zacharias, 4.22 |
| |
(2/99) |
Under "Zelinsky", add: see Eilenberg |
| |
(4/99) |
Under
"Zanardo", add: see Salce |
