Rings with a bounded number of generators for right ideals
Author:
William D. Blair
Journal:
Proc. Amer. Math. Soc. 98 (1986), 16
MSC:
Primary 16A33; Secondary 13E05, 16A38
MathSciNet review:
848862
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let the ring be a finitely generated module over a subring of its center. Then it will be shown that has the property that every right ideal can be generated by a bounded number of elements if and only if has the property that every ideal can be generated by a bounded number of elements. As a corollary we show that a twosided Noetherian affine ring satisfying a polynomial identity has the property that every right ideal can be generated by a bounded number of elements if and only if every left ideal can be generated by a bounded number of elements.
 [1]
William
D. Blair, Right Noetherian rings integral over their centers,
J. Algebra 27 (1973), 187–198. MR 0325679
(48 #4026)
 [2]
Gérard
Cauchon, Anneaux semipremiers, noethériens, à
identités polynômiales, Bull. Soc. Math. France
104 (1976), no. 1, 99–111. MR 0407076
(53 #10859)
 [3]
I.
S. Cohen, Commutative rings with restricted minimum condition,
Duke Math. J. 17 (1950), 27–42. MR 0033276
(11,413g)
 [4]
David
Eisenbud, Subrings of Artinian and Noetherian rings, Math.
Ann. 185 (1970), 247–249. MR 0262275
(41 #6885)
 [5]
Arun
Vinayak Jategaonkar, A counterexample in ring theory and
homological algebra, J. Algebra 12 (1969),
418–440. MR 0240131
(39 #1485)
 [6]
Daniel
Mollier, Descente de la propriété
noethérienne, Bull. Sci. Math. (2) 94 (1970),
25–31 (French). MR 0269638
(42 #4533)
 [7]
J.
C. Robson and Lance
W. Small, Liberal extensions, Proc. London Math. Soc. (3)
42 (1981), no. 1, 87–103. MR 602124
(82c:16025), http://dx.doi.org/10.1112/plms/s342.1.87
 [8]
Louis
Halle Rowen, Polynomial identities in ring theory, Pure and
Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace
Jovanovich, Publishers], New YorkLondon, 1980. MR 576061
(82a:16021)
 [9]
Judith
D. Sally, Some results on multiplicity with applications to bounded
and two dimensional prime bounded rings, J. Algebra
35 (1975), 224–234. MR 0379484
(52 #389)
 [10]
J.
J. Sarraillé, Module finiteness of lowdimensional PI
rings, Pacific J. Math. 102 (1982), no. 1,
189–208. MR
682051 (84f:16024)
 [11]
J.
T. Stafford, Rings with a bounded number of generators for right
ideals, Quart. J. Math. Oxford Ser. (2) 34 (1983),
no. 133, 107–114. MR 688428
(84f:16020), http://dx.doi.org/10.1093/qmath/34.1.107
 [12]
Richard
G. Swan, The number of generators of a module, Math. Z.
102 (1967), 318–322. MR 0218347
(36 #1434)
 [1]
 W. D. Blair, Right Noetherian rings integral over their centers, J. Algebra 27 (1973), 187198. MR 0325679 (48:4026)
 [2]
 G. Cauchon, Anneaux semipremiers, noethériens, à identitiés polynomiales, Bull. Soc. Math. France 104 (1976), 99111. MR 0407076 (53:10859)
 [3]
 I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950), 2742. MR 0033276 (11:413g)
 [4]
 D. Eisenbud, Subrings of Artinian and Noetherian rings, Math. Ann. 185 (1970), 247249. MR 0262275 (41:6885)
 [5]
 A. V. Jategaonkar, A counterexample in ring theory and homological algebra, J. Algebra 12 (1969), 418440. MR 0240131 (39:1485)
 [6]
 D. Mollier, Descente de la propriété noethérienne, Bull. Sci. Math. 94 (1970), 2531. MR 0269638 (42:4533)
 [7]
 J. C. Robson and L. W. Small, Liberal extensions, Proc. London Math. Soc. (3) 42 (1981), 87103. MR 602124 (82c:16025)
 [8]
 L. H. Rowen, Polynomial identities in ring theory, Academic Press, New York, 1980. MR 576061 (82a:16021)
 [9]
 J. D. Sally, Some results on multiplicity with applications to bounded and two dimensional prime bounded rings, J. Algebra 35 (1975), 224234. MR 0379484 (52:389)
 [10]
 J. J. Sarraillé, Module finiteness of lowdimensional PI rings, Pacific J. Math. 102 (1982), 189208. MR 682051 (84f:16024)
 [11]
 J. T. Stafford, Rings with a bounded number of generators for right ideals, Quart. J. Math. Oxford Ser. (2) 34 (1983), 107114. MR 688428 (84f:16020)
 [12]
 R. G. Swan, The number of generators of a module, Math. Z. 102 (1967), 318322. MR 0218347 (36:1434)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
16A33,
13E05,
16A38
Retrieve articles in all journals
with MSC:
16A33,
13E05,
16A38
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198608488620
PII:
S 00029939(1986)08488620
Article copyright:
© Copyright 1986
American Mathematical Society
