When are Baer modules extending?

The well-known notion of an extending module is closely linked to that of a Baer module. A right R-module M is called extending if every submodule of M is essential in a direct summand. On the other hand, a right R-module M is called Baer if for all N <= M, l(S)(N) <=(circle plus) SS where S = End(R)(M). In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over non-singular CS rings, J. London Math. Soc. 21(2) (1980) 434-444.] in terms of modules and showed the connections between Baer and extending modules via the result: "a module M is K-nonsingular extending if and only if M is K-cononsingular Baer". MR is called K-nonsingular if for all phi is an element of S such that Ker phi <=(e) M, phi = 0. Moreover, M-R is called K-cononsingular if for any N <= M with phi N not equal 0 for all 0 not equal phi is an element of S, implies N <=(e) M. In view of this result, every Baer module which happens to be K-cononsingular will automatically become an extending module. In this paper, our main focus is the study of K-cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of K-cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of K-cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi-Roman's result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided.