On constant products of elements in skew polynomial rings

Let R be a reversible ring which is α-compatible for an endomorphism α of R and f(X)=a0+a1X+⋯+anXn be a nonzero skew polynomial in R[X;α]. It is proved that if there exists a nonzero skew polynomial g(X)=b0+b1X+⋯+bmXm in R[X;α] such that g(X)f(X)=c is a constant in R, then b0a0=c and there exist nonzero elements a and r in R such that rf(X)=ac. In particular, r=abp for some p, 0≤p≤m, and a is either one or a product of at most m coefficients from f(X). Furthermore, if b0 is a unit in R, then a1,a2,⋯,an are all nilpotent. As an application of the above result, it is proved that if R is a weakly 2-primal ring which is α-compatible for an endomorphism α of R, then a skew polynomial f(X) in R[X;α] is a unit if and only if its constant term is a unit in R and other coefficients are all nilpoten