CS-Rickart Modules

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 circle plus 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 epsilon 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 circle plus . . . circle plus P-n circle plus 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.