On Skew Polynomials over p.q.-Baer and p.p.-Modules

Details On Skew Polynomials over p.q.-Baer and p.p.-Modules On Skew Polynomials over p.q.-Baer and p.p.-Modules

2011-01-27

arxiv.org/abs/1101.5415v2

International Mathematical Forum, Vol. 6, 2011, no. 35, 1739 - 1747

Mathematics

Rings and Algebras

Scientific paper

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 $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a p.p.-module. Many generalizations are obtained, and more results are found when $M_R$ is a semicommutative module.

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