%0 Journal Article
%T On constant products of elements in skew polynomial rings
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A Chen, W.
%D 2015
%\ 04/29/2015
%V 41
%N 2
%P 453-462
%! On constant products of elements in skew polynomial rings
%K Constant products
%K skew polynomial rings
%K reversible rings
%K weakly 2-primal rings
%R
%X Let $R$ be a reversible ring which is $\alpha$-compatible for an endomorphism $\alpha$ of $R$ and $f(X)=a_0+a_1X+\cdots+a_nX^n$ be a nonzero skew polynomial in $R[X;\alpha]$. It is proved that if there exists a nonzero skew polynomial $g(X)=b_0+b_1X+\cdots+b_mX^m$ in $R[X;\alpha]$ such that $g(X)f(X)=c$ is a constant in $R$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $R$ such that $rf(X)=ac$. In particular, $r=ab_p$ for some $p$, $0\leq p\leq m$, and $a$ is either one or a product of at most $m$ coefficients from $f(X)$. Furthermore, if $b_0$ is a unit in $R$, then $a_1,a_2,\cdots, a_n$ are all nilpotent. As an application of the above result, it is proved that if $R$ is a weakly 2-primal ring which is $\alpha$-compatible for an endomorphism $\alpha$ of $R$, then a skew polynomial $f(X)$ in $R[X;\alpha]$ is a unit if and only if its constant term is a unit in $R$ and other coefficients are all nilpotent.
%U http://bims.iranjournals.ir/article_622_9f1ac6891d2a8dd6f84d434909052144.pdf