The Schröder–Bernstein problem for dual F-Baer modules

The Baer and the quasi-Baer properties of rings are extended to a module theoretic setting in this chapter. Using the endomorphism ring of a module, the notions of Baer, quasi-Baer, and Rickart modules are introduced and studied. Similar to the fact that every Baer ring is nonsingular, we shall see that every Baer module satisfies a weaker notion of nonsingularity of modules (\(\mathcal{K}\)-nonsingularity) which depends on the endomorphism ring of the module. Strong connections between a Baer module and an extending module will be observed via this weak nonsingularity and its dual notion. It is shown that an extending module which is \(\mathcal{K}\)-nonsingular is precisely a \(\mathcal{K}\)-cononsingular Baer module. This provides a module theoretic analogue of the Chatters-Khuri theorem for rings. Direct summands of Baer and quasi-Baer modules respectively inherit these properties. This provides a rich source of examples of Baer and quasi-Baer modules, since one can readily see that for any (quasi-)Baer ring R and an idempotent e in R, the right R-module eR R is always a (quasi-)Baer module. It will be seen that every projective module over a quasi-Baer ring is a quasi-Baer module. Connections of a (quasi-)Baer module and its endomorphism ring are discussed. Characterizations of classes of rings via the Baer property of certain classes of free modules over them are presented. An application also yields a type theory for \(\mathcal{K}\)-nonsingular extending (continuous) modules which, in particular, improve the type theory for nonsingular injective modules provided by Goodearl and Boyle.

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.

Lifting modules plays important roles in module theory. H-supplemented modules are a nice generalization of lifting modules which have been studied extensively recently. In this article, we introduce a proper generalization of H-supplemented modules via images of fully invariant submodules. Let F be a fully invariant submodule of a right Rmodule M. We say that M is IF -H-supplemented in case for every endomorphism φ of M, there is a direct summand D of M such that φ(F) + X = M if and only if D + X = M, for every submodule X of M. It is proved that M is IF -H-supplemented if and only if F is a dual Rickart direct summand of M for a fully invariant noncosingular submodule F of M. It is shown that the direct sum of IF –H supplemented modules is not in general IF -H-supplemented. Some sufficient conditions such that the direct sum of IF -H-supplemented modules is IF -H-supplemented are given