Principally Quasi-Baer Skew Power Series Modules

A module M _R is called principally quasi-Baer (or simply p.q.-Baer) if the annihilator of every cyclic submodule of M _R is generated by an idempotent, as a right ideal. Let α be an automorphism of R and M _R be an α-compatible module and every countable subset of right semicentral idempotents in R has a generalized countable join or R satisfies the ACC on left annihilator ideals. It is shown that M _R is p.q.-Baer if and only if Mx]] _{Rx; α]]} is p.q.-Baer if and only if Mx, x ^?1]] _{Rx, x ^?1; α]]} is p.q.-Baer. As a consequence, we unify and extend nontrivially many of the previously known results such as 11 Hashemi , E. ( 2008 ). A note on p.q.-Baer modules . New York J. Math. 14 : 403 – 410 . Google Scholar], 15 Huang , F. K. ( 2008 ). A note on extensions of principally quasi-Baer rings . Taiwan. J. Math. 45 ( 4 ): 469 – 481 . Google Scholar], 20 Liu , Z. ( 2002 ). A note on principally quasi-Baer rings . Comm. Algebra 30 ( 8 ): 3885 – 3890 .Taylor &; Francis Online], Web of Science ®] , Google Scholar]]. Examples to illustrate and delimit the theory are provided.