Noetherian properties in composite generalized power series rings

Abstract Let ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let Γ ⁎ = Γ \ { 0 } {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D ⊆ E D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set D + 〚 E Γ ⁎ , ≤ 〛 ≔ f ∈ 〚 E Γ , ≤ 〛 | f ( 0 ) ∈ D and D + 〚 I Γ ⁎ , ≤ 〛 ≔ f ∈ 〚 D Γ , ≤ 〛 | f ( α ) ∈ I , for all α ∈ Γ ⁎ . \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D + 〚 E Γ ⁎ , ≤ 〛 D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D + 〚 E Γ ⁎ , ≤ 〛 D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D + 〚 I Γ ⁎ , ≤ 〛 D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D + ( X 1 , … , X n ) E [ X 1 , … , X n ] D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D + ( X 1 , … , X n ) I [ X 1 , … , X n ] D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

It is easy to see that 〚 〛 ≤ R Γ, is a commutative ring (under these operations) with identity e, namely, ( ) = e 0 1 and ( ) = e α 0 for every ∈ { } α Γ\ 0 , which is called the ring of generalized power series of Γ over R. , ring of formal power series over R with n indeterminates [2, Example 3]. For more details on the ring of generalized power series, the readers can refer to [1][2][3].
In [2], Ribenboim determined when a generalized power series ring 〚 〛 ≤ R Γ, is a Noetherian ring, where ( ≤) Γ, is a strictly ordered monoid. He showed that if 〚 〛 ≤ R Γ, is Noetherian, then the following three conditions hold: . Let ⊆ D E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set consists of only one element α, then we write ( ) = f α π and call it the order of f. If ( ≤) Γ, is totally ordered and ≠ ∈ f A 0 , then ( ) f supp is a nonempty wellordered subset of Γ; so ( ) f π always consists only one element.
is a Noetherian ring, then the following statements hold. (1) D is a Noetherian ring.
(2) If Γ is cancellative, then E is a finitely generated D-module and Γ is finitely generated.
Thus, E is a finitely generated D-module.
Proof of Claim. Suppose, by way of contradiction, that there is no such element contradicts our assumption and so the claim holds. Thus, assume implies, as in the argument above, that . It follows that we obtain, for every ≥ m 1, , for every ≥ m 1. Hence, for every ≥ m 1, is Noetherian, there exists an integer ≥ N 1 such that, for every ≥ k N, and Γ is cancellative, This proves the claim.
By the claim, we obtain a strictly infinite chain ⊊ ⊊⋯ I I is a Noetherian ring.

Consider the chain of ideals in
is a Noetherian ring, there exists an integer ≥ N 1 such that, for every ≥ k N, (2) If Γ is cancellative, then Γ is finitely generated.
We next prove the converse of Theorem 2.1 under some additional conditions on Γ. To do this, we need some lemmas. Hence, Γ, is a totally ordered monoid, then Γ is torsion-free and cancellative.
Γ, is an ordered monoid, and Γ is torsion-free and cancellative, then there exists a compatible strict total order on Γ, which is finer than ≤.  . We first claim So for each ∈ α S αt , we have  .) Clearly, every submonoid of an Archimedean monoid is also Archimedean; so each submonoid of m is Archimedean.
Γ, be a positively totally ordered monoid. If Γ is finitely generated, then Γ is Archimedean.
Proof. This is an immediate consequence of Lemma 2.4 and the facts that (1)   . By Lemma 2.6, ( ) = g π 0, a contradiction to the definition of g. Therefore, In general, ≼ ≼ a b a does not imply = a b; so ≼ is not always a partial order on M. We first introduce some terminology in [8]. We say that a monoid M is strict if, for every ∈ a b c M , , , + + = a b c c implies = = a b 0. For a partially ordered set ( ≤) M, , a lower set of M is a subset I of M such that, for all ∈ x y M , , ≤ ∈ x y y I , implies ∈ x I. We write ⇓( ≤) M, for the set of lower sets of M ordered by inclusion. Recall that an ordered monoid means a strictly ordered monoid. We collect some known results on a monoid M with algebraic preorder ≼ in [8]. be an increasing map between partially ordered sets. If σ is surjective and ⇓M is artinian, then ⇓N is artinian.
Thus, the identity map from ( ≼) Γ, to ( ≤) Γ, is a monoid surjection. Since ⇓( ≼) Γ, is artinian, ⇓( ≤) Γ, is artinian by Lemma 2.8 (5  Throughout this section, an ordered monoid ( ≤) Γ, means a nonzero strictly ordered monoid. Let D be a commutative ring with identity and let I be a nonzero proper ideal of D. In this section, we give an equivalent condition for the ring + 〚 〛 ≤ D I Γ , ⁎ to be Noetherian when ( ≤) Γ, is positively totally ordered. We also give some applications of composite generalized power series rings.