Classical quotient rings

Abstract

Throughout R is a ring with right singular ideal Z ( R ) Z(R) . A right ideal K of R is rationally closed if x − 1 K = { y ∈ R : x y ∈ K } {x^{ - 1}}K = \{ y \in R:xy \in K\} is not a dense right ideal for all x ∈ R − K x \in R - K . A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R / Z ( R ) R/Z(R) is semiprime, Z ( R ) Z(R) is rationally closed, and Z ( R ) 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.