A partial order on subsets of Baer bimodules with applications to C*-modules

In this paper, we introduce the concept of Baer (p, q)-sets. Using this notion, we define Rickart, Baer, quasi-Baer and pi-Baer (S, R)-bimodules, respectively. We show how these conditions relate to each other. We also develop new properties of the minus binary relation, <=-, we extend the relation <=- to (S, R)-bimodules and use it to characterize the aforementioned Rickart, Baer, quasi-Baer, and pi-Baer (S,R)-bimodules. Moreover, we specify subsets kappa of the power set of a (S,R)-bimodule for which <=- determines a partial order and for which <=- is a lattice. We analyze the relation <=- by examining the associated Baer (p, q)-sets. Finally, we apply our results to C*-modules. Examples are provided to illustrate and delimit our results.