When is the maximal ring of quotients projective?

Handelman, David

doi:10.1090/S0002-9939-1975-0389955-7

When is the maximal ring of quotients projective?
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by David Handelman

Proc. Amer. Math. Soc. 52 (1975), 125-130

DOI: https://doi.org/10.1090/S0002-9939-1975-0389955-7

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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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