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 \({\varvec{R}}\) -modules of T.K. Lee and Y. Zhou and the theory of Baer annihilator conditions for \((\textbf{H}, {\varvec{R}})\) -bimodules of G. Lee, S.T. Rizvi, and C.S. Roman where \(\textbf{H}= {\varvec{End}}({\varvec{M}}_{\varvec{R}})\) and M is a right \(\varvec{R}\) -module. To encompass the theory of Baer annihilator conditions for \((\textbf{H}, \varvec{R})\) -bimodules of Lee, Rizvi, and Roman, we consider Baer annihilator conditions for \({\varvec{(S, R)}}\) -bimodules where \(\varvec{S}\) may not be \(\textbf{H}\) . One of the major pioneering results of the \((\textbf{H}, {\varvec{R}})\) -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 \({\varvec{(S, R)}}\) -bimodules where \({\varvec{S}}\) is not restricted to being \(\textbf{H}\) . Among other results, we investigate conditions on \({\varvec{S}}\) or a left \({\varvec{S}}\) -module, \({\varvec{M}}\) , such that either one or both satisfy a Baer annihilator condition. Examples are provided to illustrate and delimit our results.
