On The Hilbert Basis Theorem Over Skew Polynomial Rings and Skew Power Series Rings

Let R be a unitary ring and  : RR be a ring endomorphism of R . In this note, we will provide another proof of Hilbert basis Theorem for skew polynomial ring R[x; ] and skew power series ring R[[x; ]]. Also, we will extend this theorem for rings [ ; 1, 2,..., n] n R F    and [[ ; 1, 2,..., n ]] n R F    such that F be a free monoid and 1,2,...,n are ring automorphisms.