Extensions of Baer and quasi-Baer modules. (Q648460)

The author investigates the relationships between the Baer, quasi-Baer and p.q.-Baer property of an \(R\)-module \(M\) and the polynomial extensions of the module \(M\). He shows that if \(M_R\) is an \((\alpha,\delta)\)-compatible module, then \(M_R\) is quasi-Baer (resp. p.q.-Baer) if and only if \(M[x]_S\) is quasi-Baer (resp. p.q.-Baer); in this case, \(M_R\) is an \((\alpha,\delta)\)-quasi Armendariz module. By three examples the author justifies that the \(\alpha\)-compatibility condition on \(M_R\) in the previous result is not superfluous. The author finds some results of \textit{E. Hashemi} and \textit{A. Moussavi} [Acta Math. Hung. 107, No. 3, 207-224 (2005; Zbl 1081.16032)] and \textit{C. Y. Hong, N. K. Kim} and \textit{T. K. Kwak} [J. Pure Appl. Algebra 151, No. 3, 215-226 (2000; Zbl 0982.16021)] as a consequence of his results. Finally the author shows that if \(S=R[x;\alpha,\delta]\) and \(M_R\) is an \((\alpha,\delta)\)-compatible Armendariz module then \(M_R\) is Baer (resp. p.p.) if and only if \(M[x]_S\) is Baer (resp. p.p.).