The extended centralizer of a ring over a module

In a recent paper,1 K. Asano gave a new proof of the theorem that a domain of integrity has a right quotient ring if and only if every pair of nonzero elements has a common nonzero right multiple.His method of proof is used in the present work to extend the centralizer of a ring over a module to a system of semi-endomorphisms of the module.From this extension, necessary and sufficient conditions that a ring have a right quotient regular ring are derived.Consider a given ring R, and a given nonzero right i?-moduIe M.Denote by 3JÎ the set of all submodules of M, and by SDÎ* the set of all submodules N oí M having the property that NÍ^N't^O for all nonzero TV'GSOî-Since ME^R*, SDÎ* is not void.It is easily seen that if N and N' are in M*, then N+N' and NC\N' are also in Stt*.Thus {9ÏÏ*; Q, r\, +} is a sublattice of the lattice {W; Q, C\, + }.An i?-homomorphism of N into M, N any element of SO?, is called a semi-endomorphism of M. Thus, thinking of the semi-endomorphism a as a left operator on N, we have a(x+y) =ax+ay and a(xa) = (ax)a for all x, yEN, aER-For convenience, the module N on which a is