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Transactions of the American Mathematical Society

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



Classical quotient rings

Author: Robert C. Shock
Journal: Trans. Amer. Math. Soc. 190 (1974), 43-48
MSC: Primary 16A08
MathSciNet review: 0338044
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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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Keywords: Classical quotient ring, complete ring of right quotients, injective, cogenerator, Artinian, quasi-Frobenius ring, rationally closed right ideal
Article copyright: © Copyright 1974 American Mathematical Society