Dual CS-Rickart Modules over Dedekind Domains

Abstract

We study d-CS-Rickart modules (i.e. modules M such that for every endomorphism φ of M, the image of φ lies above a direct summand of M) over Dedekind domains. The structure of d-CS-Rickart modules over discrete valuation rings is fully determined. It is also shown that for a d-CS-Rickart R-module M over a nonlocal Dedekind domain R, the following assertions hold:(i)The p-primary component of M is a direct summand of M for any nonzero prime ideal p of R.(ii)M/T(M) is an injective R-module, where T(M) is the torsion submodule of M.(iii)If, moreover, M is a reduced R-module, then ⊕p∈PTp(M)≤M≤∏p∈PTp(M), where P is the set of all nonzero prime ideals of R and Tp(M) is the p-primary component of M for every p∈ P.