On constant products of elements in skew polynomial rings

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 $p$, $0leq pleq m$, 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.