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 $\mathfrak {p}$-primary component of M is a direct summand of M for any nonzero prime ideal $\mathfrak {p}$ 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 $\bigoplus _{\mathfrak {p} \in \mathbf {P}} T_{\mathfrak {p}}(M) \leq M \leq {\prod }_{\mathfrak {p} \in \mathbf {P}} T_{\mathfrak {p}}(M),$ where P is the set of all nonzero prime ideals of R and $T_{\mathfrak {p}}(M)$ is the $\mathfrak {p}$-primary component of M for every $\mathfrak {p} \in \mathbf {P}$.