Polynomial extensions of modules with the quasi-Baer property

In this paper it is shown that, for a module M over a ring R with S = End(R)(M), the endomorphism ring of the R[x]-module M[x] is isomorphic to a subring of S[[x]]. Also the endomorphism ring of the Rp[x]]-module M[[x]]( )is isomorphic to S[[x]]. As a consequence, we show that for a module M-R and an arbitrary nonempty set of not necessarily commuting indeterminates X, M-R is quasi-Baer if and only if M[X](R[x] )is quasi-Baer if and only if M[[X]](R[[x]]) is quasi-Baer if and only if M[x](R[x]) is quasi-Baer if and only if M[[x]](R[[x]]) is quasi-Baer. Moreover, a module M-R with IFP, is Baer if and only if M[x](R[x]) is Baer if and only if M[[x]](R)([[)(x)(]]) is Baer. It is also shown that, when MR is a finitely generated module, and every semicentral idempotent in S is central, then M[[X]](R[[X]]) is endo-p.q.-Baer if and only if M[[x]](R[[x]]) is endo-p.q.-Baer if and only if M-R is endo-p.q.-Baer and every countable family of fully invariant direct summand of M has a generalized countable join. Our results extend several existing results. (C) 2019 Elsevier Inc. All rights reserved.