On skew polynomials over p.q.-Baer and p.p.-modules. (English) Zbl 1250.16023

Louzari, Mohamed

Int. Math. Forum 6, No. 33-36, 1739-1747 (2011).

Summary: Let \(M_R\) be a module and \(\sigma\) an endomorphism of \(R\). Let \(m\in M\) and \(a\in R\), we say that \(M_R\) satisfies the condition \(\mathcal C_1\) (respectively, \(\mathcal C_2\)), if \(ma=0\) implies \(m\sigma(a)=0\) (respectively, \(m\sigma(a)=0\) implies \(ma=0\)).
We show that if \(M_R\) is p.q.-Baer then so is \(M[x;\sigma]_{R[x;\sigma]}\) whenever \(M_R\) satisfies the condition \(\mathcal C_2\), and the converse holds when \(M_R\) satisfies the condition \(\mathcal C_1\). Also, if \(M_R\) satisfies \(\mathcal C_2\) and \(\sigma\)-skew Armendariz, then \(M_R\) is a p.p.-module if and only if \(M[x;\sigma]_{R[x;\sigma]}\) is a p.p.-module if and only if \(M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}\) (\(\sigma\in\operatorname{Aut}(R)\)) is a p.p.-module. Many generalizations are obtained and more results are found when \(M_R\) is a semicommutative module.

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16W20	Automorphisms and endomorphisms
16P60	Chain conditions on annihilators and summands: Goldie-type conditions
16D40	Free, projective, and flat modules and ideals in associative algebras
16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)

Keywords:

semicommutative modules; p.q.-Baer modules; pp-modules