| 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/
Gårding |
|
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) |
