Chinese Journal of Contemporary Mathematics (Chinese J. Contemp. Math. ) (1999), no.~2, 255--260 ISSN: 19411146, 08985111. eISSN: 1941-1146.
Subject
16 Associative rings and algebras -- 16D Modules, bimodules and ideals 16D70 Structure and classification
Language
Chinese
Online Access
초록
Let $R$ be a left semi-Artinian ring (= left socular ring), $\phi$ the set of representations of the isomorphism classes of its simple modules. The main result in the paper is that the conditions below are equivalent. (1) $R$ is primary decomposable; (2) $|\phi|$ is finite and ${\rm Ext}_R(S_1,S^{(A)}_2)=0$ for any set $A$ of indices and non-isomorphic simple modules $S_1,S_2$; (3) $|\phi|$ is finite, ${\rm Ext}_R(S_1,S^{(A)}_2)=0$ for any set $A$ of indices and non-isomorphic simple modules $S_1,S_2$. And the primitive ideal is always maximal; (4) $R$ is right perfect and $P_1P_2=P_2P_1$ for any maximal ideals $P_1,P_2$ [cf. J. S. Alin, Math. Z. {\bf 107} (1968), 319--325; MR0238898 (39 \#258); S. E. Dickson, Math. Z. {\bf 104} (1968), 349--357; MR0229678 (37 \#5252)].