On ‎Constant ‎Products Of Elements ‎In ‎Skew ‎t.u.p. ‎Monoid ‎Rings

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and omega : M→ End(R)a monoid homomorphism. Let R be a reversible, M-compatible ring and alpha=a_{1}g_{1}+...+a_{n}g_{n} a non-zero element in skew monoid ring R*M. It is proved that if there exists a non-zero element beta=b_{1}h_{1}+...+b_{m}h_{m} in R*M such that alpha beta=c is a constant, then there exist 1≤ i_{0} ≤n, 1 ≤j_{0}≤ m such that g_{i_{0}}=e=h_{j_{0}} and a_{i_{0}}b_{j_{0}}=c and there exist elements a, 0 ̸= r in R with alpha r=ca.