Dual CS-Rickart Modules over Dedekind Domains

  • Rachid Tribak


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:
  1. (i)

    The \(\mathfrak {p}\)-primary component of M is a direct summand of M for any nonzero prime ideal \(\mathfrak {p}\) of R.

  2. (ii)

    M/T(M) is an injective R-module, where T(M) is the torsion submodule of M.

  3. (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 \(\mathfrak {p} \in \mathbf {P}\).



Endomorphisms d-CS-Rickart modules d-Rickart modules Lifting modules 

Mathematics Subject Classification (2010)

16D10 16D70 16D80 


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The author is very grateful to the referee for valuable suggestions and comments which improved this paper.


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Authors and Affiliations

  1. 1.(CRMEF)-TangerCentre Régional des Métiers de L’Education et de la Formation (CRMEFTTH)TangierMorocco

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