Endoregular Modules

We study modules whose endomorphism rings possess various forms of von Neumann regularity. We characterize these "regularity" properties for several classes of modules, including completely decomposable modules and finitely presented modules over commutative rings. We generalize many of the classic results about abelian groups with regular endomorphism rings to modules over one-dimensional commutative rings with Noetherian spectrum. To facilitate this study, we define a module M over a commutative ring R to be weakly endoregular if xM and (0:M(x)) are direct summands of M for each x∈R. We give several characterizations of weak endoregularity for various classes of modules.