Baer, Quasi-Baer Modules, and Their Applications

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. Similar to the case of Baer modules, close links between quasi-Baer modules and FI-extending modules are established via a characterization connecting the two notions. The concepts of FI- $\mathcal{K}$ -nonsingularity and FI- $\mathcal{K}$ -cononsingularity are introduced and utilized to obtain this characterization. Analogous to right Rickart rings, the notion of Rickart modules is introduced as another application of the theory of Baer modules in the last section of the chapter. Connections of Rickart modules to their endomorphism rings are shown. A direct sum of Rickart modules is not Rickart in general. The closure of the class of Rickart modules with respect to direct sums is discussed among other recent results on this notion.