Noetherian rings of composite generalized power series

Let A. B be an extension of commutative rings with identity, S =,() a nonzero strictly ordered monoid, and S = S * \ 0{}. Let A + B = = f. B = f. A 0S S *,,.. {.. | ()}. In this study, we determine when the ring A + BS = *,.. is a Noetherian ring. We prove that when S is a strict monoid, if A + BS = *,.. is a Noetherian ring, then A is a Noetherian ring, B is a finitely generated A-module, and S is finitely generated. We also show that if B is a finitely generated A-module over a Noetherian ring A and S =,() is a positive strictly ordered monoid, which is finitely generated, then A + BS = *,.. is a Noetherian ring.