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?
HTML articles powered by AMS MathViewer

by David Handelman PDF

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 $R \ne Q$.

References

Efraim P. Armendariz, A note on semiprime rings with torsionless injective envelope, Canad. Math. Bull. 16 (1973), 429–431. MR 338046, DOI 10.4153/CMB-1973-067-7
Carl Faith, Lectures on injective modules and quotient rings, Lecture Notes in Mathematics, No. 49, Springer-Verlag, Berlin-New York, 1967. MR 0227206
K. R. Goodearl and D. Handelman, Simple self-injective rings, Comm. Algebra 3 (1975), no. 9, 797–834. MR 379593, DOI 10.1080/00927877508822074
David Handelman and John Lawrence, Strongly prime rings, Trans. Amer. Math. Soc. 211 (1975), 209–223. MR 387332, DOI 10.1090/S0002-9947-1975-0387332-0

Strongly prime, completely torsion-free and simple self-injective rings

Joachim Lambek, Lectures on rings and modules, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. With an appendix by Ian G. Connell. MR 0206032
Joachim Lambek, Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, Vol. 177, Springer-Verlag, Berlin-New York, 1971. With an appendix by H. H. Storrer on torsion theories and dominant dimensions. MR 0284459
Robert A. Rubin, Absolutely torsion-free rings, Pacific J. Math. 46 (1973), 503–514. MR 364326

On absolutely torsion-free rings and kernel functors

Vasily C. Cateforis, On regular self-injective rings, Pacific J. Math. 30 (1969), 39–45. MR 248178