Co-Artinian rings and Morita duality |
| |
| Authors: | Robert W Miller Darrell R Turnidge |
| |
| Institution: | (1) Department of Mathematics, College of William and Mary, Williamsburg, Virginia, U.S.A.;(2) Department of Mathematics, Kent State University, Kent, Ohio, U.S.A. |
| |
| Abstract: | A ringA is left co-Noetherian if the injective hull of each simple leftA-module is Artinian. Such rings have been studied by Vámos and Jans. Dually, callA left co-Artinian if the injective hull of each simple leftA-module is Noetherian. Left co-Artinian rings having only finitely many nonisomorphic simple left modules are studied, and
such rings are shown to have nilpotent radical. Moreover, it is shown that left co-Artinian implies left co-Noetherian ifA/J is Artinian. For an injective leftA-module
A
Q withB=End (
A
Q), andC=End (Q
B
), conditions yielding a Morita duality between
and
are obtained. In special cases, e.g.
A
Q a self-cogenerator, this Morita duality yields chain conditions on
A
Q. Specialized to commutative rings, these results give the known fact that every commutative co-Artinian ring is co-Noetherian.
Finally in the case that the injective hull
A
E=E(
A
S) of a simple leftA-module
A
S is a self-cogenerator, chain conditions on
A
E are related to chain conditions onB
B
=End (
A
E). The results obtained are analogous to results for commutative rings of Vámos, Rosenberg and Zelinsky. It is shown that
ifA is a left co-Artinian ring withE(
A
S) a self-cogenerator for each simple
A
S, thenJ is nil and
. |
| |
| Keywords: | |
| 本文献已被 SpringerLink 等数据库收录！ |
|
