Regular Modules

Abstract

In analogy to the elementwise definition of von Neumann regular rings an $R$-module $M$ is called regular if given any element $m \in M$ there exists $f \in {\operatorname {Hom} _R}(M,R)$ with $(mf)m = m$. Other equivalent definitions are possible, and the basic properties of regular modules are developed. These are applied to yield several characterizations of regular self-injective rings. The endomorphism ring $E(M)$ of a regular module $_RM$ is examined. It is in general a semiprime ring with a regular center. An immediate consequence of this is the recently observed fact that the endomorphism ring of an ideal of a commutative regular ring is again a commutative regular ring. Certain distinguished subrings of $E(M)$ are also studied. For example, the ideal of $E(M)$ consisting of the endomorphisms with finite-dimensional range is a regular ring, and is simple when the socle of $_RM$ is homogeneous. Finally, the self-injectivity of $E(M)$ is shown to depend on the quasi-injectivity of $_RM$.