Orders in self-injective cogenerator rings

Abstract:

This note states necessary and sufficient conditions for a ring to be a right order in certain self-injective rings. A ring R is said to have the dense extension property if each R-homomorphism from a right ideal of R into R can be lifted to some dense right ideal of R. A right ideal K is rationally closed if for each $x \in R - K$ the set ${x^{ - 1}}K = \{ y \in R:xy \in K\}$ is not a dense right ideal of R. We state a major result. Let $\dim R$ denote the Goldie dimension of a ring R and $Z(R)$ the right singular ideal of R. Then R is a right order in a self-injective cogenerator ring if and only if R has the dense extension property, $Z(R)$ is rationally closed and the factor ring $R/Z(R)$ is semiprime with $\dim R/Z(R) = \dim R < \infty$.