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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Orders in self-injective cogenerator rings

Author: Robert C. Shock
Journal: Proc. Amer. Math. Soc. 35 (1972), 393-398
MSC: Primary 16A18
MathSciNet review: 0302683
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Abstract: This note states necessary and sufficient conditions for a ring to be a right order in certain self-injective rings. A ring R is said to have the dense extension property if each R-homomorphism from a right ideal of R into R can be lifted to some dense right ideal of R. A right ideal K is rationally closed if for each $ x \in R - K$ the set $ {x^{ - 1}}K = \{ y \in R:xy \in K\} $ is not a dense right ideal of R. We state a major result. Let $ \dim \, R$ denote the Goldie dimension of a ring R and $ Z(R)$ the right singular ideal of R. Then R is a right order in a self-injective cogenerator ring if and only if R has the dense extension property, $ Z(R)$ is rationally closed and the factor ring $ R/Z(R)$ is semiprime with $ \dim \, R/Z(R) = \dim R < \infty $.

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Keywords: Injective, order, cogenerator, rationally closed, quasi-Frobenius, dense right ideal, complete ring of right quotients
Article copyright: © Copyright 1972 American Mathematical Society

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