Given an R-module M, the centralizer near-ring a?3R (M) is the set of all functions f: M a?’ M with f(xr)= f(x)r for all x a?? M and ra??R endowed with point-wise addition and composition of functions as multiplication. In general, a?3R(M) is not a ring but is a near-ring containing the endomorphism ring ER(M) of M. Necessary and/or sufficient conditions are derived for a?3R(M) to be a ring. For the case that R is a Dedekind domain, the R-modules M are characterized for which (i) a?3R(M) is a ring; and (ii)a?3R(M) = ER(M). It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.
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