Rings with a bounded number of generators for right ideals

Blair, William D.

doi:10.1090/S0002-9939-1986-0848862-0

Rings with a bounded number of generators for right ideals
HTML articles powered by AMS MathViewer

by William D. Blair

Proc. Amer. Math. Soc. 98 (1986), 1-6

DOI: https://doi.org/10.1090/S0002-9939-1986-0848862-0

PDF | Request permission

Abstract:

Let the ring $S$ be a finitely generated module over a subring $R$ of its center. Then it will be shown that $S$ has the property that every right ideal can be generated by a bounded number of elements if and only if $R$ has the property that every ideal can be generated by a bounded number of elements. As a corollary we show that a two-sided 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.

References

William D. Blair, Right Noetherian rings integral over their centers, J. Algebra 27 (1973), 187–198. MR 325679, DOI 10.1016/0021-8693(73)90173-7
Gérard Cauchon, Anneaux semi-premiers, noethériens, à identités polynômiales, Bull. Soc. Math. France 104 (1976), no. 1, 99–111. MR 407076
I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950), 27–42. MR 33276
David Eisenbud, Subrings of Artinian and Noetherian rings, Math. Ann. 185 (1970), 247–249. MR 262275, DOI 10.1007/BF01350264
Arun Vinayak Jategaonkar, A counter-example in ring theory and homological algebra, J. Algebra 12 (1969), 418–440. MR 240131, DOI 10.1016/0021-8693(69)90040-4
Daniel Mollier, Descente de la propriété noethérienne, Bull. Sci. Math. (2) 94 (1970), 25–31 (French). MR 269638
J. C. Robson and Lance W. Small, Liberal extensions, Proc. London Math. Soc. (3) 42 (1981), no. 1, 87–103. MR 602124, DOI 10.1112/plms/s3-42.1.87
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
Judith D. Sally, Some results on multiplicity with applications to bounded and two dimensional prime bounded rings, J. Algebra 35 (1975), 224–234. MR 379484, DOI 10.1016/0021-8693(75)90047-2
J. J. Sarraillé, Module finiteness of low-dimensional PI rings, Pacific J. Math. 102 (1982), no. 1, 189–208. MR 682051
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, DOI 10.1093/qmath/34.1.107
Richard G. Swan, The number of generators of a module, Math. Z. 102 (1967), 318–322. MR 218347, DOI 10.1007/BF01110912