# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Classical quotient ringsHTML articles powered by AMS MathViewer

by Robert C. Shock
Trans. Amer. Math. Soc. 190 (1974), 43-48 Request permission

## Abstract:

Throughout R is a ring with right singular ideal $Z(R)$. A right ideal K of R is rationally closed if ${x^{ - 1}}K = \{ y \in R:xy \in K\}$ is not a dense right ideal for all $x \in R - K$. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, $R/Z(R)$ is semiprime, $Z(R)$ is rationally closed, and $Z(R)$ contains no closed uniform right ideals. These rings include the quasi-Frobenius rings as well as the semiprime Goldie rings. The commutative Cl-rings have Cl-classical quotient rings. The injective ones are congenerator rings. In what follows, R is a Cl-ring. A dense right ideal of R contains a right nonzero divisor. If R satisfies the minimum condition on rationally closed right ideals then R has a classical Artinian quotient ring. The complete right quotient ring Q (also called the Johnson-Utumi maximal quotient ring) of R is a Cl-ring. If R has the additional property that bR is dense whenever b is a right nonzero divisor, then Q is classical. If Q is injective, then Q is classical.
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