## Classical quotient rings

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- by Robert C. Shock PDF
- 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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## Additional Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**190**(1974), 43-48 - MSC: Primary 16A08
- DOI: https://doi.org/10.1090/S0002-9947-1974-0338044-X
- MathSciNet review: 0338044