Centralizer near-rings that are rings

Abstract

Abstract Given an R -module M , the centralizer near-ring ℳ R ( M ) is the set of all functions f: M → M with f(xr)= f(x)r for all x ∈ M and r∈R endowed with point-wise addition and composition of functions as multiplication. In general, ℳ R (M) is not a ring but is a near-ring containing the endomorphism ring E R (M) of M . Necessary and/or sufficient conditions are derived for ℳ R (M) to be a ring. For the case that R is a Dedekind domain, the R -modules M are characterized for which (i) ℳ R (M) is a ring; and (ii) ℳ R (M) = E R (M) . It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.