Noetherian Generalized Power Series Rings#

Abstract

Let R be a unitary ring and (M, ≤) a strictly ordered monoid. We show that, if (M, ≤) is positively ordered, then the generalized power series ring R[[M, ≤]] is left Noetherian, if and only if, R is left Noetherian and M is finitely generated, if and only if, R is left Noetherian and R[[M, ≤]] is a homomorphic image of the power series ring R[[x 1, x 2,…, x n ]] for some n ∈ ℕ.