Localizations of HNP rings

Kuzmanovich, James

doi:10.1090/S0002-9947-1972-0311699-X

Localizations of HNP rings
HTML articles powered by AMS MathViewer

by James Kuzmanovich PDF

Trans. Amer. Math. Soc. 173 (1972), 137-157 Request permission

Abstract:

In this paper it is shown that every hereditary Noetherian prime ring is the intersection of a hereditary Noetherian prime ring having no invertible ideals with a bounded hereditary Noetherian prime ring in which every nonzero two-sided ideal contains an invertible two-sided ideal. Further, it is shown that this intersection corresponds to a decomposition of torsion modules over such a ring; if $R$ is an HNP ring with enough invertible ideals, then this decomposition coincides with that of Eisenbud and Robson. If $M$ is a maximal invertible ideal of $R$ where $R$ is as above, then an overring of $R$ is constructed which is a localization of $R$ at $M$ in a “classical sense"; that is, it is a ring of quotients with respect to a multiplicatively closed set of regular elements satisfying the Ore conditions. The localizations are shown to have nonzero radical and are also shown to satisfy a globalization theorem. These localizations are generalizations of ones constructed by A. V. Jategaonkar for HNP rings with enough invertible ideals.

References

Commuting elements in free algebras and related topics in ring theory

Spencer E. Dickson, A torsion theory for Abelian categories, Trans. Amer. Math. Soc. 121 (1966), 223–235. MR 191935, DOI 10.1090/S0002-9947-1966-0191935-0
David Eisenbud and J. C. Robson, Hereditary Noetherian prime rings, J. Algebra 16 (1970), 86–104. MR 291222, DOI 10.1016/0021-8693(70)90042-6
David Eisenbud and J. C. Robson, Modules over Dedekind prime rings, J. Algebra 16 (1970), 67–85. MR 289559, DOI 10.1016/0021-8693(70)90041-4
L. Fuchs, Torsion preradicals and ascending Loewy series of modules, J. Reine Angew. Math. 239(240) (1969), 169–179. MR 285565, DOI 10.1515/crll.1969.239-240.169
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821, DOI 10.24033/bsmf.1583
Oscar Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10–47. MR 245608, DOI 10.1016/0021-8693(69)90004-0
Nathan Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys, Vol. II, American Mathematical Society, New York, 1943. MR 0008601, DOI 10.1090/surv/002
James Kuzmanovich, Localizations of Dedekind prime rings, J. Algebra 21 (1972), 378–393. MR 311698, DOI 10.1016/0021-8693(72)90002-6
Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on torsion theories and dominant dimensions. MR 0284459, DOI 10.1007/BFb0061029
Lawrence Levy, Torsion-free and divisible modules over non-integral-domains, Canadian J. Math. 15 (1963), 132–151. MR 142586, DOI 10.4153/CJM-1963-016-1
J.-M. Maranda, Injective structures, Trans. Amer. Math. Soc. 110 (1964), 98–135. MR 163937, DOI 10.1090/S0002-9947-1964-0163937-X
J. C. Robson, Idealizers and hereditary Noetherian prime rings, J. Algebra 22 (1972), 45–81. MR 299639, DOI 10.1016/0021-8693(72)90104-4
J. C. Robson, Non-commutative Dedekind rings, J. Algebra 9 (1968), 249–265. MR 231849, DOI 10.1016/0021-8693(68)90024-0
Carol L. Walker and Elbert A. Walker, Quotient categories and rings of quotients, Rocky Mountain J. Math. 2 (1972), no. 4, 513–555. MR 338045, DOI 10.1216/RMJ-1972-2-4-513
D. B. Webber, Ideals and modules of simple Noetherian hereditary rings, J. Algebra 16 (1970), 239–242. MR 265395, DOI 10.1016/0021-8693(70)90029-3