Akademik Veri Yönetim Sistemi
LOBACHEVSKII JOURNAL OF MATHEMATICS, cilt.40, sa.12, ss.2103-2110, 2019 (ESCI)
We introduce a symmetry property for unit-regular rings as follows: a is an element of R is unit-regular if and only if aR circle plus (a - u)R = R (equivalently, Ra circle plus R(a - u) = R) for some unit u of R if and only if aR circle plus (a - u)R =(2a - u)R (equivalently, Ra circle plus R(a - u) = R(2a - u)) for some unit u of R. Let M and N be right R-modules and alpha, beta is an element of Hom(M, N) such that alpha + beta is regular. It is shown that alpha S circle plus beta S =(alpha + beta)S, where S = End(M) if and only if T alpha circle plus T beta = T(alpha + beta), where T = End(N). We also introduce partial order alpha <=(circle plus)beta and minus partial order alpha <=(-)beta for any alpha, beta is an element of Hom(M, N); they translate into module-theoretic language defined in a ring in [7] and [8]. We analyze some relationships between <=(circle plus) and <=(-) on the endomorphism rings of the modules M and N.