ABSTRACT
We define “regular” for maps in a Hom group. This notion specializes to the well-known notions of (Von Neumann) regular in rings and modules. A map f ∈ Hom R (A,M) is regular if and only if Ker(f) ⊆⊕ A and Im(f) ⊆⊕ M. There exists a unique maximal regular End(M)-End(A)-submodule in Hom R (A,M). We study regularity in Hom R (A 1 ⊕ A 2, M 1 ⊕ M 2). The existence of a regular function Hom R (A,M) implies the existence of projective summands of Hom R (A,M) End R (A) and of End R ( M ) Hom R (A,M). We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.
Mathematics Subject Classification:
Notes
Communicated by K. M. Rangaswamy.