CENTRALIZER NEAR-RINGS THAT ARE RINGS

Given an R-module M, the centralizer near-ring M(R)(M) is the set of a ll functions f : M --> M with f(xr) = f(x)r for all x is an element of M and r is an element of R endowed with point-wise addition and compo sition of functions as multiplication. Tn general, M(R) (M) is not a r ing 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 c haracterized 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.