Generalized Power Series Rings Research Articles

Noetherian Generalized Power Series Rings#

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.

On t-closedness of generalized power series rings

For an extension A⊆ B of commutative rings, we present a sufficient condition for the ring [[ A S,⩽ ]] of generalized power series to be t-closed in [[ B S,⩽ ]], where ( S,⩽) is a torsion-free cancellative ordered monoid. As a corollary, this result can be applied to the ring of power series in any number of indeterminates.

Krull Domains of Generalized Power Series

For an integral domain D and a torsion-free cancellative strictly subtotally ordered monoid (S,≤), it is shown that the generalized power series ring [[DS,≤]] is a Krull domain if and only if D is a Krull domain and S is a Krull monoid.

On n-root closedness of generalized power series rings over pairs of rings

Given commutative rings A⊆ B, we present a sufficient condition for the generalized power series ring [[ A S,≤ ]] to be n-root closed in [[ B S,≤ ]], where ( S,≤) is a torsion-free, cancellative and strictly ordered monoid.