On (Unit-)Regular Morphisms

Abstract

We introduce a symmetry property for unit-regular rings as follows: a ∈ R is unit-regular if and only if aR ⊕ (a − u)R = R (equivalently, Ra ⊕ R(a − u) = R) for some unit u of R if and only if aR ⊕ (a − u)R =(2a − u)R (equivalently, Ra ⊕ R(a − u) = R(2a − u)) for some unit u of R. Let M and N be right R-modules and α, β ∈ Hom(M, N) such that α + β is regular. It is shown that αS ⊕ βS =(α + β)S, where S = End(M) if and only if Tα ⊕ Tβ = T(α + β), where T = End(N). We also introduce partial order α ≤^⊕β and minus partial order α ≤⁻β for any α, β ∈ Hom(M, N); they translate into module-theoretic language defined in a ring in [7] and [8]. We analyze some relationships between ≤^⊕ and ≤⁻ on the endomorphism rings of the modules M and N.