For an endomorphism alpha of R, in , a module M-R is called alpha-compatible if, for any m is an element of M and a is an element of R, ma = 0 iff m alpha(a) = 0, which are a generalization of alpha-reduced modules. We study on the relationship between the quasi-Baerness and p.q.-Baer property of a module MR and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of  and some results in . In particular, we show: for an alpha-compatible module M-R (1) M-R is p.q.-Baer module iff M[x;alpha] R-[x,R-alpha] is p.q.-Baer module. (2) for an automorphism alpha of R, MR is p.q.-Baer module iff M[x, x(-1); alpha](R[x,x-1;alpha,]) is p.q.Baer module.