# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## When is the maximal ring of quotients projective?HTML articles powered by AMS MathViewer

by David Handelman
Proc. Amer. Math. Soc. 52 (1975), 125-130 Request permission

## 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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