When are Baer modules extending?

Abstract

The well-known notion of an extending module is closely linked to that of a Baer module. A right [Formula: see text]-module [Formula: see text] is called extending if every submodule of [Formula: see text] is essential in a direct summand. On the other hand, a right [Formula: see text]-module [Formula: see text] is called Baer if for all [Formula: see text], [Formula: see text] where [Formula: see text]. In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular 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[Formula: see text] is[Formula: see text]-nonsingular extending if and only if[Formula: see text] is[Formula: see text]-cononsingular Baer”. [Formula: see text] is called [Formula: see text]-nonsingular if [Formula: see text] such that [Formula: see text], [Formula: see text]. Moreover, [Formula: see text] is called [Formula: see text]-cononsingular if for any [Formula: see text] with [Formula: see text] for all [Formula: see text], implies [Formula: see text]. In view of this result, every Baer module which happens to be [Formula: see text]-cononsingular will automatically become an extending module. In this paper, our main focus is the study of [Formula: see text]-cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of [Formula: see text]-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 [Formula: see text]-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.