Regularity and Substructures of Hom

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.

Key Words:

• Cosingular submodule
• Maximal regular bi-submodule
• Partially invertible
• Projective summand
• Radical
• Regular homomorphism
• Singular sub-module
• Total
• Von Neumann regular

Mathematics Subject Classification:

• Primary 16D10, 16D80
• Secondary 20K25, 20K30

Notes

Communicated by K. M. Rangaswamy.