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Mediterranean Journal of Mathematics, vol.22, no.8, 2025 (SCI-Expanded)
In this paper, we provide a framework which enables us to abstract and extend various Baer, quasi-Baer, Rickart, and p.q.-Baer conditions (i.e., Baer annihilator conditions) for modules. In particular, this framework allows us to generalize the theory of Baer annihilator conditions for right R-modules of T.K. Lee and Y. Zhou and the theory of Baer annihilator conditions for (H,R)-bimodules of G. Lee, S.T. Rizvi, and C.S. Roman where H=End(MR) and M is a right R-module. To encompass the theory of Baer annihilator conditions for (H,R)-bimodules of Lee, Rizvi, and Roman, we consider Baer annihilator conditions for (S,R)-bimodules where S may not be H. One of the major pioneering results of the (H,R)-bimodule theory by Rizvi and Roman was to obtain a module analogue of the Chatters–Khuri Theorem which links the Baer condition and the extending condition for rings. Our theory generalizes the Rizvi–Roman result to (S,R)-bimodules where S is not restricted to being H. Among other results, we investigate conditions on S or a left S-module, M, such that either one or both satisfy a Baer annihilator condition. Examples are provided to illustrate and delimit our results.