GPQ modules and generalized Armendariz modules

Let MR be a right R-module. We introduce the concept of right generalized p.q.-Baer modules (or simply, right GPQ modules) to extend the notion of right p.q.-Baer modules. We study on the relationship between the GPQ property of a module MR and various quasi-Armendariz properties. We prove that every right GPQ module is a quasi-Armendariz module. As a sequence, we obtain a general form of some known results considering the p.q.Baer property of a ring, some known results are extended. Moreover, we prove that for the formal triangular ring R constructed from a pair of rings S, T and a bimodule SMT , R is weak Armendariz if and only if (1) S and T are weak Armendariz rings. (2) SM and MT are weak Armendariz as a left S -module and right R-module. (3) If s(x)s (x)= t(x)t (x)=0, then s(x)M[x] cap M[x]t (x)=0. This gives the relationship of weak Armendarizness between R and S, T,S MT, which plays a very important role in ring theory.