In this paper, we investigate some properties of SIP, SSP and CS-Rickart modules. We give equivalent conditions for SIP and SSP modules; establish connections between the class of semisimple artinian rings and the class of SIP rings. It shows that R is a semisimple artinian ring if and only if RR is SIP and every right R-module has a SIP-cover. We also prove that R is a semiregular ring and J(R) = Z(RR) if only if every finitely generated projective module is a SIP-CS module which is also a C2 module.2010 Mathematics Subject Classification. 16D40, 16D80.

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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.

The classical reverse 1-median problem on trees is to adjust the edge lengths within a budget so as to reduce the 1-median function at a predetermined vertex as much as possible. This paper concerns the corresponding problem with uncertain vertex weights presented by linear functions. Moreover, we use the minmax regret criterion to measure the maximum loss of a feasible solution with respect to the worst-case scenario. The regarding problem is called the minmax regret reverse 1-median problem on trees. We first partition the set of scenarios into parts such that the optimal solution of the corresponding reverse 1-median problem does not change in each part. Then the problem can be reformulated as the minimization of a quadratic number of affine linear functions. We finally develop a greedy algorithm that solves the problem in [Formula: see text] time where [Formula: see text] is the number of vertices in the underlying tree.

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