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.References
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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