Abstract
In this paper, we introduce and study the concept of CS-Rickart modules, that is a module analogue of the concept of ACS rings. A ring R is called a right weakly semihereditary ring if every its finitly generated right ideal is of the form P ⊕ S, where P R is a projective module and S R is a singular module. We describe the ring R over which Mat n (R) is a right ACS ring for any n ∈ N. We show that every finitely generated projective right R-module will to be a CS-Rickart module, is precisely when R is a right weakly semihereditary ring. Also, we prove that if R is a right weakly semihereditary ring, then every finitely generated submodule of a projective right R-module has the form P 1 ⊕ … ⊕ P n ⊕ S, where every P 1, …, P n is a projective module which is isomorphic to a submodule of R R , and S R is a singular module. As corollaries we obtain some well-known properties of Rickart modules and semihereditary rings.
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Abyzov, A.N., Nhan, T.H.N. CS-Rickart modules. Lobachevskii J Math 35, 317–326 (2014). https://doi.org/10.1134/S199508021404009X
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DOI: https://doi.org/10.1134/S199508021404009X