Mohamed Louzari
On McCoy Condition and Semicommutative Rings.
Wed, 04/25/2012 - 15:01 — SomNambulAuthors: Mohamed Louzari
Subjects: Rings and Algebras
AbstractLet $R$ be a ring, $\sigma$ an endomorphism of $R$, $I$ a right ideal in
$S=R[x;\sigma]$ and $M_R$ a right $R$-module. We give a generalization of
McCoy's Theorem \cite{mccoy}, by showing that, if $r_S(I)$ is $\sigma$-stable
or $\sigma$-compatible. Then $\;r_S(I)\neq 0$ implies $r_R(I)\neq 0$. As a
consequence, if $R[x;\sigma]$ is semicommutative then $R$ is $\sigma$-skew
McCoy. Moreover, we show that the Nagata extension $R\oplus_{\sigma}M_R$ is
semicommutative right McCoy when $R$ is a commutative domain.On skew polynomials over p.q.-Baer and p.p.-modules.
Mon, 01/31/2011 - 16:01 — SomNambulAuthors: Mohamed Louzari
Subjects: Rings and Algebras
AbstractLet $M_R$ 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 $\mathcal{C}_1$.
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