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Proceedings of the American Mathematical Society

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

 

 

When is the maximal ring of quotients projective?


Author: David Handelman
Journal: Proc. Amer. Math. Soc. 52 (1975), 125-130
MSC: Primary 16A08
MathSciNet review: 0389955
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Abstract: Let $ R$ be an associative ring with $ 1$, and $ Q$ its maximal ring of right quotients. If $ r$ belongs to $ R$, a right insulator for $ r$ in $ R$ is a finite subset of $ R,\{ {r_i}\} _{i = 1}^m$, such that the right annihilator of $ \{ r{r_i};i = 1, \ldots ,m\} $ is zero. Then we have: If $ Q$ is a projective right $ R$-module, $ Q$ is finitely generated; if $ R$ is nonsingular, then $ Q$ is projective as a right $ R$-module if and only if there exists $ e = {e^2}$ in $ R$ such that $ eR$ is injective and $ e$ has a right insulator in $ R$; under these circumstances, $ R = Q$ if and only if $ e$ has a left insulator in $ R$. We prove some related results for torsionless $ Q$, and give an example of a prime ring $ R$ such that $ Q$ is a cyclic projective right $ R$-module, but $ R \ne Q$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0389955-7
Keywords: Ring of quotients, projective, injective, torsionless, nonsingular, prime ring, regular ring, simple ring
Article copyright: © Copyright 1975 American Mathematical Society