ON CONSTANT PRODUCTS OF ELEMENTS IN SKEW POLYNOMIAL RINGS

Let R be a reversible ring which is a-compatible for an endomorphism alpha of R and f(X) = a(0) + a(1)X + ... + a(n) X-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(1)X + ... + b(m)X(m) in R[X; alpha] such that 9(X) f (X) = c is a constant in R, then b(0)a(0) = c and there exist nonzero elements a and r in R such that r f(X) = ac. In particular, r = ab(p) for some p, 0 <= p <= 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), ..., 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 a-compatible for an endomorphism a 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.