Noetherian Properties in Composite Generalized Power Series Rings

Open Mathematics 2020; 18: 1540–1551

Research Article

Jung Wook Lim and Dong Yeol Oh* Noetherian properties in composite generalized power series rings

https://doi.org/10.1515/math-2020-0103 received March 7, 2020; accepted October 20, 2020

Abstract: Let (Γ, ≤) be a strictly ordered monoid, and let Γ⁎ ={Γ\ 0}. Let D ⊆ E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set

⁎ DE+〚Γ,≤≤ 〛≔{ fE ∈〚 Γ, 〛| f (0and )∈ D } ⁎ DI+〚Γ,≤≤ 〛≔{ fD ∈〚 Γ, 〛| fαI ( )∈,forall α ∈ Γ. ⁎ }

⁎ In this paper, we give necessary conditions for the rings DE+〚 Γ,≤〛to be Noetherian when(Γ, ≤) is positively ⁎ ordered, and suﬃcient conditions for the rings DE+〚 Γ,≤〛 to be Noetherian when (Γ, ≤) is positively totally ⁎ ordered. Moreover, we give a necessary and suﬃcient condition for the ring DI+〚 Γ,≤〛 to be Noetherian when (Γ, ≤) is positively totally ordered. As corollaries, we give equivalent conditions for the rings

DX+(11,, … XEXXnn ) [ ,, … ] and DX+(11,, … XIXXnn )[ ,, … ] to be Noetherian.

⁎⁎ Keywords: DE+〚ΓΓ,,≤≤ 〛, DI +〚〛, generalized power series ring, Noetherian ring

MSC 2020: 13A02, 13A15, 13B35, 13E05

1 Introduction

Throughout this paper, a monoid means a commutative semigroup with identity element. The operation is written additively and the identity element is denoted by 0, unless otherwise stated. A monoid Γ is said to be cancellative if every element in Γ is cancellative, i.e., for every α,,βγ∈ Γ, α +=+βαγimplies β = γ. We denote by G(Γ) the largest subgroup of Γ, i.e., G()≔{ΓΓααβ ∈ | + = 0,for someβ ∈ Γ}. A monoid Γ is torsion-free if for an α ∈ Γ and a positive integer n, nα = 0 implies α = 0.If n α1,,…∈αn Γ, then we denote by 〈αα1,,… n 〉 the set of all elements ∑i=1 kαiiwith nonnegative integers ki.A monoid Γ is finitely generated if there exists a finite subset {αα1,,… n} of Γ such that Γ =〈αα1,, … n 〉.Anideal of Γ is a nonempty subset I of Γ such that IαI⊇+≔{+|∈ αγγI} for each α ∈ Γ. An ordered monoid is a monoid (Γ, +) together with a partial order ≤ that is compatible with the monoid operation, meaning that for every α12,,αβ∈ Γ, α1 ≤ α2 implies α12+≤βα +β. An ordered monoid (Γ, ≤) is positively ordered if 0 ≤ α for all α ∈ Γ. Note that if (Γ, ≤) is positively ordered, then G()={Γ0}. An ordered monoid (Γ, ≤) is a strictly ordered monoid if for every α12,,αβ∈ Γ, α1 < α2 implies α12+<βα +β. We say that an ordered monoid (Γ, ≤) is artinian if every decreasing sequence of elements of Γ is finite; and (Γ, ≤) is narrow if every subset of pairwise order-incomparable elements of Γ is finite.

 * Corresponding author: Dong Yeol Oh, Department of Mathematics Education, Chosun University, Gwangju 61452, Republic of Korea, e-mail: [email protected], [email protected] Jung Wook Lim: Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea, e-mail: [email protected]

Open Access. © 2020 Jung Wook Lim and Dong Yeol Oh, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. Noetherian properties in composite generalized power series rings  1541

Let R be a commutative ring with identity and(Γ, ≤) a strictly ordered monoid. We denote by〚〛RΓ,≤ the set of all mappings f :Γ→ R such that supp()≔{∈fαfα Γ |()≠ 0} is an artinian and narrow subset of Γ. (The set supp( f ) is called the support of f.) With pointwise addition,〚〛RΓ,≤ is an (additive) abelian group. Moreover, Γ,≤ for every α ∈ Γ and f , gR∈〚〛,thesetXα()≔{()∈×|=+()≠fg,,ΓΓ, βγ α β γfβ 0,andgγ()≠0} is ﬁnite [1,1.16]; so this allows us to deﬁne the operation of convolution ⁎ on 〚〛RΓ,≤ : For every f , gR∈〚 Γ,≤〛, ( fg⁎ )( α ) =∑ fβgγ ( ) ( ). ()∈()βγ,, Xα f g

It is easy to see that 〚〛RΓ,≤ 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. The elements of 〚〛RΓ,≤ are called generalized power series with coefficients in R and exponents in Γ.Itis well known that R is canonically embedded as a subring of 〚〛RΓ,≤ and Γ is canonically embedded as Γ,≤ Γ,≤ a submonoid of 〚R 〛{\0} by the mapping α ∈↦Γ eRα ∈〚〛, where eα()=α 1 and eα()=γ 0 for every Γ,≤ α - γα∈{Γ\ }. The elements of 〚〛R are written in the form ∑αf∈()supp fαX() , with addition and multiplica Γ,≤ α tion defined as for formal power series. Thus, a mapping eα ∈〚R 〛 is written in the form X . Henceforth, α Γ,≤ we use the notation ∑αf∈()supp fαX() for the elements of 〚〛R . Let be the additive monoid of nonnegative integers with the usual order ≤. Then every subset of is artinian and narrow; thus 〚RRX,≤〛≅ [ ], ring of formal power series over R in one indeterminate X [2, Example 2].If is the usual ordered monoid, and Γ ≔=×⋯×n (n times) with the order ≤, where ≤ is the product order, or the lexicographic order, or the reverse lexicographic order, then Γ,≤ 〚RRXX〛≅ [1,, … n], 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 [1–3]. In [2], Ribenboim determined when a generalized power series ring 〚〛RΓ,≤ is a Noetherian ring, where (Γ, ≤) is a strictly ordered monoid. He showed that if 〚〛RΓ,≤ is Noetherian, then the following three conditions hold: (1) R is Noetherian; (2) if Γ is cancellative, then there exist α1,,…∈αGn Γ\Γ () such that Γ ⊆〈αα1,, …n 〉+ G ( Γ); and (3) if 0 ≤ α for every α ∈ Γ, then (Γ, ≤) is narrow [2, 5.2]. He also proved the converse under an additional hypothesis which states that 〚〛RΓ,≤ is Noetherian if R and Γ satisfy the following: (1) (Γ, ≤) is narrow and Γ is torsion-free cancellative; (2) there exist α1,,…∈αGn Γ\Γ () such that Γ =〈αα1,, …n 〉+ G ( Γ); and (3) R is Noetherian [2, 5.5].In[4, Theorem 4], Hizem and Benhissi showed that if D ⊆ E is an extension of commutative rings, then DXEX+[] is a Noetherian ring if and only if D is a Noetherian ring and E is a finitely generated D-module. In [5, Proposition 2.4], Hizem proved that DXIX+[] is a Noetherian ring if and only if D is a Noetherian ring and II= 2. Let (Γ, ≤) be a strictly ordered monoid, and let Γ⁎ ≔{Γ\ 0}. Let D ⊆ E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set

⁎ DE+〚Γ,≤≤ 〛≔{ fE ∈〚 Γ, 〛| f (0and )∈ D } ⁎ DI+〚Γ,≤≤ 〛≔{ fD ∈〚 Γ, 〛| fαI ( )∈,forall α ∈ Γ. ⁎ }

⁎ ⁎ It is easy to see that DE+〚 Γ,≤〛 and DI+〚 Γ,≤〛 are commutative rings with identity, which are called the ⁎⁎ composite generalized power series rings. Then DD⊊ +〚 IΓ,≤≤ 〛⊊〚 D Γ, 〛⊆ D +〚 E Γ, ≤≤ 〛⊆〚 E Γ, 〛. Note that if G()≠{Γ0}, then for 0Γ≠∈(αG), there exists β ∈ Γ such that α +=β 0. For aE∈ , a =∈+aXαβ⁎ X D ⁎ ⁎ 〚〛EΓ,≤ . Thus, DE+〚Γ,≤≤ 〛=〚 E Γ, 〛. Hence, if E properly contains D and (Γ, ≤) is positively ordered, then ⁎ DE+〚 Γ,≤〛 is properly between 〚〛DΓ,≤ and 〚〛EΓ,≤ . We note that if Γ = n with the product order, or the ⁎ ⁎ lexicographic order, or the reverse lexicographic order, then DE+〚 Γ,≤〛 and DI+〚 Γ,≤〛 are isomorphic to

DX+(1,, … Xn)EX[…1,, Xn] and DX+(11,, … XIXXnn )[ ,, … ], respectively. The composite generalized power series rings are also appropriate examples of D + M constructions. ⁎ Also, DE+〚 Γ,≤〛 guarantees some algebraic properties of intermediate rings between〚〛DΓ,≤ and〚〛EΓ,≤ , and ⁎ DI+〚 Γ,≤〛 provides us some information of generalized power series rings which are contained in the usual generalized power series ring 〚〛DΓ,≤ . Let (Γ, ≤) be a strictly ordered monoid, and let Γ⁎ ≔{Γ\ 0}. Let D ⊆ E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. In this article, we study Noetherian properties 1542  Jung Wook Lim and Dong Yeol Oh

⁎ ⁎ on the composite generalized power series rings DE+〚 Γ,≤〛 and DI+〚 Γ,≤〛. In Section 2, we determine ⁎ when the ring DE+〚 Γ,≤〛 is a Noetherian ring. More precisely, we give necessary conditions for the ring ⁎ DE+〚 Γ,≤〛 to be Noetherian when (Γ, ≤) is positively ordered, and sufficient conditions for the ring ⁎ DE+〚 Γ,≤〛 to be Noetherian when (Γ, ≤) is positively totally ordered. As a corollary, if (Γ, ≤) is positively ⁎ totally ordered, then DE+〚 Γ,≤〛 is a Noetherian ring if and only if D is a Noetherian ring, E is a finitely generated D-module, and Γ is finitely generated. Thus, we recover [4, Theorem 4] that DXEX+[] is a Noetherian ring if and only if D is a Noetherian ring and E is a finitely generated D-module. In Section 3, ⁎ when (Γ, ≤) is positively totally ordered, we show that DI+〚 Γ,≤〛 is a Noetherian ring if and only if D is Noetherian, II= 2, and Γ is finitely generated. Thus, we recover [5, Proposition 2.4] that DXIX+[] is a Noetherian ring if and only if D is a Noetherian ring and II= 2.

2 Composite generalized power series ring of the form D +[[EΓ⁎,≤]]

Throughout this section, an ordered monoid (Γ, ≤) means a nonzero strictly ordered monoid. Let (Γ, ≤) be a positively ordered monoid, and Γ⁎ ={Γ\ 0}. Let D ⊆ E be an extension of commutative ⁎ rings with identity. In this section, we determine when the ring DE+〚 Γ,≤〛 is a Noetherian ring. Γ,≤ Γ,⁎ ≤ Let A be either〚〛E or DE+〚〛. For simplicity, the ideal of A generated by f1,,… fn of A is denoted by ( ff1,,… n) or ( ff1,,…)n A instead of ( ff1,,…)n ⁎A, and the principal ideal of A generated by fA∈ is denoted by fA instead of fA⁎ . Note that for everyf , g ∈ A, supp(fgfg + )⊆ supp ( )∪ supp ( ) and supp ( fgfg ⁎ )⊆ supp ( )+ supp ( ).

For 0 ≠∈f A, we denote by π( f ) the set of minimal elements in supp( f ). Then π( f ) is a nonempty ﬁnite set consisting of pairwise order incomparable elements. If π( f ) consists of only one element α, then we write π()=f α and call it the order of f.If(Γ, ≤) is totally ordered and 0 ≠∈f A, then supp( f ) is a nonempty well- ordered subset of Γ;soπ( f ) always consists only one element. ⁎ We now give necessary conditions for the ring DE+〚 Γ,≤〛 to be Noetherian. For the most part of the proof, we follow the proof of [2, 5.2].

Theorem 2.1. Let D ⊆ E be an extension of commutative rings with identity and (Γ, ≤) a positively ordered ⁎ monoid. If DE+〚 Γ,≤〛 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. (3) (Γ, ≤) is narrow.

Proof. ⁎ ⁎ ⁎⁎ (1) Note that 〚〛EΓ,≤ is an ideal of DE+〚 Γ,≤〛, and DD≅( +〚 EΓ,≤≤ 〛)/〚 E Γ, 〛;soD is a Noetherian ring.

(2) Let Γ be cancellative. We ﬁrst show that E is ﬁnitely generated as a D-module. Let α ∈ Γ⁎. Consider the ideal Xα〚EΓ,≤〛 of 〚〛EΓ,≤ generated by Xα:

Xαα〚〛≔{|∈〚〛}EXggEΓ,≤≤⁎ Γ, .

⁎ ⁎ Note that α ∈ Γ⁎, so for every gE∈〚 Γ,≤〛, Xα⁎gD∈+〚 EΓ,≤〛. Thus, Xα〚EΓ,≤〛 is an ideal of DE+〚 Γ,≤〛. Γ,⁎ ≤ α Γ,≤≤Γ,⁎ α Γ,≤ Since DE+〚〛 is Noetherian, X 〚〛=(…)(+〚〛EggDE1,,n ), for some ggXE1,,…∈〚n 〛. Then, for Γ,≤ α some fi ∈〚E 〛, each gXfi ≔ ⁎ i. Hence, we have

ααΓ,≤≤Γ,⁎⁎⁎⁎ α Γ, ≤α Γ, ≤ Γ, ≤ XE〚〛 = XfDE⁎⁎⁎1 ( +〚〛)+⋯+ XfDEn ( +〚〛)= XfDE (1 ( +〚〛)+⋯+ fDEn ( +〚〛) .

Since Γ is cancellative, Ef=()+⋯+()1 00 D fn D. Thus, E is a ﬁnitely generated D-module. Noetherian properties in composite generalized power series rings  1543

Next, we prove that Γ is ﬁnitely generated. Suppose that Γ ≠〈αα1,, … n 〉 for any ﬁnite subset {αα1,,… n} of Γ⁎. For each n ≥ 1, set

ααΓ,⁎⁎≤≤ Γ, IXDEn ≔1 ( +〚〛)+⋯+ XDEn ( +〚〛).

α Claim: There exists an element αnn+11∈〈…Γ\αα , , 〉 such that X n+1 ∉ In.

Proof of Claim. Suppose, by way of contradiction, that there is no such element αn+1. Let β ∈〈…Γ\αα1 , , n 〉. β β If X ∉ In, then αn+1 ≔ β contradicts our assumption and so the claim holds. Thus, assume X ∈ In. Then ⁎ Γ,≤ β n αi there exist f1,,…∈+〚fDEn 〛 such that X =∑i=1 Xf⁎ i. Hence, we have  n  n n β suppXfαii ⁎ supp Xfα ⁎ α supp f ; ∈⊆⋃()=⋃(+())∑  i ii i=1 i  i==11i so β =+αγi1 1 for some in1 ∈{1, … , } and γf1 ∈(supp i1 ). Note that γα1 ∉〈1,, … αn 〉 because β ∉〈αα1,, … n 〉.If γ X 1 ∉ In, then setting αn+1 ≔ γ1 again contradicts our assumption because γαα1 ∉<1,, …n >. Thus, assume γ X 1 ∈ In. We proceed by induction. Assume m ≥ 1 and that γβγγ01=…∈,,,m Γ and ii1 …∈{…,1,,m n} have been chosen so that, for each 1 ≤≤j m,

γj γαj ∉〈1,, … αn 〉 , XIγαγγ ∈n ,j−1 =ijj +jj , ∈ supp ( fi ).

γ γ Note that for β =∉〈…γγ00,,, α1 αn 〉 and X 0 ∈ In. Then X m ∈ In implies, as in the argument above, that

γαm =+im+1 γm+1, for some inm+1 ∈{1, 2, … , } and some γfααm+1 ∈()〈…suppinm+1 \1 , , 〉. By the contradiction γ hypothesis, X m+1 ∈ In. It follows that we obtain, for every m ≥ 1,

βαα=++⋯++=++⋯++ii12 α immm γααm ii 12 α i α i+ 1 + γm+1.

Since Γ is cancellative, γαm =+im+1 γm+1, for every m ≥ 1. Hence, for every m ≥ 1,

γα γ X m = XXim+1 ⁎ m+1, ⁎ and so we have an inﬁnite nondecreasing chain of ideals in DE+〚 Γ,≤〛:

γγγΓ,⁎⁎ Γ, Γ,⁎ X 1 (+〚DE≤≤≤ 〛)⊆⋯⊆ XDEmm (+〚〛)⊆ X+1 (+〚 DE 〛)⊆⋯. ⁎ Since DE+〚 Γ,≤〛 is Noetherian, there exists an integer N ≥ 1 such that, for every kN≥ ,

⁎⁎ X γγkN(+〚DEΓ,≤≤ 〛)= XDE (+〚 Γ, 〛).

⁎ γγN 1 N Γ,≤ Therefore, X + = Xf⁎ , for some f ∈+〚DE〛 and thus γγδNN+1 =+, for some δ ∈(supp f ). Since

γαN =+iN+1 γN+1 and Γ is cancellative, αiN+1 +=δ 0. Since 0 ≤ α for all α ∈ Γ, G()={Γ0};soαiN+1 = 0, which ⁎ contradicts the fact that αi ∈ Γ . This proves the claim.

Γ,⁎ ≤ By the claim, we obtain a strictly inﬁnite chain II12⊊⊊⋯of ideals in DE+〚〛, which is a contra- ⁎ diction to the fact that DE+〚 Γ,≤〛 is a Noetherian ring.

(3) Suppose to the contrary that there are inﬁnitely many pairwise incomparable elements α12,,α … of Γ. ⁎ Consider the chain of ideals in DE+〚 Γ,≤〛:

⁎ ⁎ ⁎ X ααα1 (+〚DEΓ,≤≤≤ 〛)⊆ XDE1 (+〚 Γ, 〛)+ XDE2 (+〚 Γ, 〛)⊆⋯.

⁎ Since DE+〚 Γ,≤〛 is a Noetherian ring, there exists an integer N ≥ 1 such that, for every kN≥ ,

⁎⁎⁎⁎ X αααα1 (+〚DEΓ,≤≤≤≤ 〛)+⋯+ XDEk (+〚 Γ, 〛)= XDE1 (+〚 Γ, 〛)+⋯+ XDEN (+〚 Γ, 〛).

N ⁎ αN+1 αi Γ,≤ Therefore, X =∑i=1 Xf⁎ i, for some f1,,…∈+〚fDEN 〛. Hence, we obtain  N  N N αiiα αn+1 ∈⊆⋃()=⋃(+())supp∑Xf ⁎i  supp Xf ⁎i αii supp f. i=1  i==11i 1544  Jung Wook Lim and Dong Yeol Oh

Thus, there exist kN∈{1, … , } and β ∈(supp fk ) such that αNk+1 =+α β, which implies that αNk+1 ≥ α , a contradiction. □

Note that if (Γ, ≤) is positively ordered, then G()={Γ0}. By applying Theorem 2.1 to the case of D = E,we have the following which is the same as [2, 5.2] under the condition that (Γ, ≤) is positively ordered.

Corollary 2.2. (cf. [2, 5.2]) Let D be a commutative ring with identity and (Γ, ≤) a positively ordered monoid. If 〚〛DΓ,≤ is a Noetherian ring, then the following statements hold. (1) D is a Noetherian ring. (2) If Γ is cancellative, then Γ is ﬁnitely generated. (3) (Γ, ≤) is narrow.

We next prove the converse of Theorem 2.1 under some additional conditions on Γ. To do this, we need some lemmas.

Lemma 2.3. Let D ⊆ E be an extension of commutative rings with identity and (Γ, ≤) an ordered monoid. Then ⁎ ⁎ ⁎ an ideal of DE+〚 Γ,≤〛 containing 〚〛EΓ,≤ is of the form IE+〚 Γ,≤〛 for some ideal I of D.

⁎ ⁎ Proof. Let A be an ideal of DE+〚 Γ,≤〛 containing 〚〛EΓ,≤ and set IffA≔{ (0 )|∈ }. Clearly, I is an ideal of D. ⁎ Choose dI∈ . Then there exists an element fA∈ such that d =(f 0). Since f −∈〚dEΓ,≤〛, dA∈ ;soI ⊆ A. ⁎ Hence, IE+〚Γ,≤ 〛⊆A. The reverse containment is obvious, which completes the proof. □

Let (Γ, ≤) and (Γ, ≤′) be ordered monoids. We say that ≤′ is ﬁner than ≤ if for every α,Γβ ∈ , αβ≤ implies αβ≤′ .

Lemma 2.4. [2, 3.2, 3.3] Let (Γ, ≤) be an ordered monoid. Then the following statements hold: (1) If (Γ, ≤) is a totally ordered monoid, then Γ is torsion-free and cancellative. (2) If (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 ≤.

Lemma 2.5. Let D ⊆ E be an extension of commutative rings with identity, and I be a finitely generated ideal of D. Let (Γ, ≤) be a positively totally ordered monoid. If E is a finitely generated D-module and Γ is finitely ⁎ generated, then IE+〚 Γ,≤〛 is finitely generated.

Proof. Note that if (Γ, ≤) is a totally ordered monoid, then by Lemma 2.4, Γ is torsion-free and cancellative.

Let EeD=+⋯+1 eDm for some e1,,…∈em E, and Γ =〈αα1,, … n 〉 with 0 <<<⋯<αα12 αn.Wefirst claim ⁎ ⁎ ⁎ that 〚〛EΓ,≤ is a finitely generated ideal of DE+〚 Γ,≤〛. Let f ∈〚EΓ,≤〛.Define

n   Sα1 ≔∈αfαkαksupp () =ii and1 ≠ 0 ,  ∑   i=1  and for in=…1, , −1, set

n  i  Sαij+1 ≔∈αfSαkαksupp ()⋃ = \α ∑ jj and i+1 ≠ 0.    j=1 ji=+1 

n Then supp()=⋃fSi=1 αi and hence we write f as follows: For 1 ≤≤tn, n α Γ,⁎ ≤ f ==∈〚〛∑∑ffaXEtt, where α . t=∈1 αSαt Noetherian properties in composite generalized power series rings  1545

Note that for each α ∈ Sαt,

aedαα=+⋯+11 ed mαm, for some d ααm 1 , …∈ , d D ,  n αkα, for some nonnegative integers kk , , with k 1.  =…≥∑ ii t n t  it=

So for each α ∈ Sαt, we have n α ∑ kαii αt αα−−t αt ααt aXedα =(11αmαm +⋯+ edX )it= =( eXdX 1 )⁎⁎. (α 1 )+⋯+( eXdXm ) (αm )

Note that if α ∈ Sαt , then α −∈αt Γ. Since (Γ, ≤) is compatible, we have that if α, βS∈ αt with αβ≠ ,

α −≠−αβαt t. We also note that {αααS−|∈tαt} is artinian because supp( f ) is artinian, which means ⁎ that ∑ XDEαα−≤t ∈+〚Γ, 〛. αS∈ αt Therefore, we have

α faXt = ∑ α αS∈ αt n ∑ kαii =(∑ ed11αmαm +⋯+) e d X it= αS∈ αt α αα− α αα− =+⋯+eX11t ⁎⁎∑∑ dα Xtt em X dαm X t αS∈ αt αS∈ αt α Γ,⁎⁎≤≤α Γ, ∈eXDE1 tt ( +〚〛)+⋯+ eXDEm ( +〚〛),

n αt and hence f =∑t=1 feXiti ∈({ | =1, … , mand tn=…}1, , ). Thus, we obtain

Γ,⁎ ≤ α 〚EeXimtn〛=({i t | =1, … , and = 1, … , }).

If Ib=(1,, … bq), then we have

Γ,⁎ ≤ α IE+〚〛=({ beXiij,k | = 1,,, … qj = 1,,,and … m k = 1,, … n }) ,

⁎ and thus IE+〚 Γ,≤〛 is ﬁnitely generated as desired. □

⁎ Recall that an additive semigroup Γ is Archimedean if ⋂n≥1(+)=∅nα Γ for each α ∈ Γ . A simple example of an Archimedean semigroup is m, where is the monoid of nonnegative integers and m is a positive m⁎ m integer. (To see this, for any (aa1,,…)∈m ,if(bb111,,…)∈⋂((…)+mn≥ naa ,, m ),thenbi ≥ nai for all im=…1, , and all n ≥ 1. However, this is impossible, because at least one of ai is nonzero. Thus, m ⋂nm≥11((na,, … a )+ )=∅.) Clearly, every submonoid of an Archimedean monoid is also Archimedean; so each submonoid of m is Archimedean.

Lemma 2.6. Let (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) S is a torsion-free, cancellative, ﬁnitely generated monoid with G()={S 0} if and only if S is isomorphic to a submonoid of m for some integer m ≥ 1 [6, Theorem 3.11](or [7, II. 7]); and (2) every submonoid of m is Archimedean. □

Let R be a commutative ring with identity and I a nonzero proper ideal of R. It is well known that by n taking {I }n≥1 to be a fundamental system of neighborhoods of 0 in R, R becomes a topological ring and the topology of R is called the I-adic topology on R. Note that the I-adic topology on R is Hausdorﬀ if and only n if ⋂n≥1 I =(0).

Lemma 2.7. Let D ⊆ E be an extension of commutative rings with identity and (Γ, ≤) a positively totally ordered ααΓ,≤≤ Γ, Γ,⁎ ≤ monoid. If Γ is generated by α1,,… αn, then the (XE1 〚〛+⋯+〚〛 XEn )-adic topology on DE+〚〛 is Hausdorﬀ. 1546  Jung Wook Lim and Dong Yeol Oh

αα1 Γ,≤≤n Γ, m Proof. Let 0 ≠∈⋂(gXEXEm≥1 〚〛+⋯+〚〛). Then for each m ≥ 1, g is a ﬁnite sum of generalized kαmmnn11+⋯+ kα Γ,≤ power series of the form X ⁎gm, where gEm ∈〚〛 and kkmmn1,,… are nonnegative integers with kkmmn1 +⋯+ = m. Recall that π(g) is the minimal element in supp(g). Therefore, we have

  πsuppkαmmnn11+⋯+ kα ⁎ supp, ()∈gXgkαkαg ∑ m  ⊆⋃(mmnn11 +⋯+ + (m )) finite  finite that is, ππ()=gkαmmnn11 +⋯+ kα + ( gm), where kmi are nonnegative integers such that kkmmn1 +⋯+ = m Γ,≤ and gEm ∈〚〛. Note that as m approaches ∞, at least one of the kmi, say kmr, for some r with 1 ≤≤rn, gets ∞ arbitrarily large. Thus, by replacing the sequence {kmr}m=1 with a subsequence if necessary, we may assume that for each m ≥ 1, we have kkmr< m+1 r and kmr ≥ m. Now

ππΓ()=gkαmmrrmnn11 +⋯+ kα +⋯+ kα + ( gm )∈ ⇒

π,forΓ()=gkαγmr r +mm γ ∈ ⇒ πΓ.()∈gmαr +

Thus, πΓ()∈⋂gmαmr≥1 ( + ). By Lemma 2.6, π()=g 0, a contradiction to the deﬁnition of g. Therefore, ααm1 Γ,≤≤n Γ, ⋂m≥1 (〚XE 〛+⋯+〚 XE 〛)=(0). □

Let M be a monoid. The algebraic preorder (or natural preorder) on a monoid M is the relation ≼ deﬁned as follows: For every a, bM∈ , a ≼+=∈bacbcMif and only if , for some .

In general, a ≼≼b a does not imply a = b;so≼ is not always a partial order on M.Weﬁrst introduce some terminology in [8]. We say that a monoid M is strict if, for every a,,bc∈ M, a ++=bcc implies a ==b 0. For a partially ordered set (M, ≤),alower set of M is a subset I of M such that, for all x, y ∈ M, x ≤∈yy, I implies xI∈ . 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].

Lemma 2.8. Let M be a nonzero monoid. Then the following statements hold: (1)[8, Lemma 3.1] If (M, ≤) is a positively ordered monoid, then M is strict. (2)[8, Lemma 3.2] M is strict if and only if (M, ≼) is an ordered monoid. (3)[8, Lemma 3.3] Let M be a strict monoid. Then M is ﬁnitely generated if and only if ⇓(≼M, ) is artinian. (4)[8, Lemma 2.2] Let (M, ≤) be a partially ordered set. Then ⇓(≤M, ) is artinian if and only if (M, ≤) is artinian and narrow. (5)[8, Lemma 2.1 (2)] Let σ : MN→ be an increasing map between partially ordered sets. If σ is surjective and ⇓M is artinian, then ⇓N is artinian.

Lemma 2.9. Let (Γ, ≤) be a positively totally ordered monoid. If Γ is ﬁnitely generated, then (Γ, ≤) is artinian.

Proof. Let (Γ, ≤) be a positively totally ordered monoid that is ﬁnitely generated. By Lemma 2.8(1), (2), and (3), (Γ, ≼) is an ordered monoid and ⇓(≼Γ, ) is artinian. Since ≤ is a positive order, we obtain α ≼⇒≤βαβ,forall, αβ ∈ Γ.

Thus, the identity map from (Γ, ≼) to (Γ, ≤) is a monoid surjection. Since ⇓(≼Γ, ) is artinian, ⇓(≤Γ, ) is artinian by Lemma 2.8(5). Hence, (Γ, ≤) is artinian by Lemma 2.8(4). □

⁎ Now we are ready to give suﬃcient conditions for DE+〚 Γ,≤〛 to be Noetherian. For simplicity, we use ⁎ the notation f n =⋯ff⁎⁎(n times) for f ∈+〚DEΓ,≤〛. Noetherian properties in composite generalized power series rings  1547

Theorem 2.10. If E is a ﬁnitely generated D-module over a Noetherian ring D and (Γ, ≤) is a positively totally ⁎ ordered monoid that is ﬁnitely generated, then DE+〚 Γ,≤〛 is a Noetherian ring.

⁎ Proof. By the hypothesis, there exist e1,,…∈en E and α1,,…∈αm Γsuch that

EeD=+⋯+11 eDnmand Γ =〈…〉 α , , α.

⁎ Suppose to the contrary that DE+〚 Γ,≤〛 is not a Noetherian ring. Let be the set of non-finitely generated Γ,⁎ ≤ ideals of DE+〚〛. Then is nonempty. Let {Bkk} ∈Λ be a chain of members of , where Λ is an indexed set. Clearly, ⋃k∈Λ Bk is an upper bound of {Bkk} ∈Λ and is non-finitely generated. By Zorn’s lemma, there exists ⁎ an ideal I of DE+〚 Γ,≤〛 that is maximal among non-finitely generated ideals. Note that I is a prime ideal of ⁎ ⁎ ⁎ DE+〚 Γ,≤〛 [9, Theorem 7].If〚EΓ,≤〛⊆I, then II=()+〚0 EΓ,≤〛, where IffI()={()|∈00} by Lemma 2.3. Since D is a Noetherian ring, I(0) is finitely generated; so I is also finitely generated by Lemma 2.5, which Γ,⁎ ≤ α contradicts the choice of I. Therefore, 〚E 〛⊈I. Hence, ei X j ∉ I for some ei and αj. Without loss of gene- α α Γ,⁎ ≤ α Γ,⁎ ≤ rality, we mayassume thate1X 1 ∉ I;soIIeXD⊊+1 1 (+〚 E 〛). By the maximality of I, IeXD+(+〚〛1 1 E ) Γ,⁎ ≤ is a finitely generated ideal of DE+〚〛; so there exists a finitely generated subideal Jf≔(1,, … fq) of I such ααΓ,⁎ ≤≤Γ,⁎ that IeXD+(+〚〛)=+(+〚〛1 1 E JeXD1 1 E ). α α Γ,⁎ ≤ α We first show that IJeX=+1 1 I. Let fI∈ . Then f ∈+JeXD1 1 (+〚 E 〛);sof =+geX1 1 ⁎h for some Γ,⁎ ≤ α Γ,⁎ ≤ α g ∈ J and hD∈+〚 E〛. Hence, e1Xhfg1 ⁎ =− ∈I. Since I is a prime ideal of DE+〚〛 and e1X 1 ∉ I,we α have h ∈ I. Therefore, f ∈+JeX1 1 I. The reverse containment is obvious. α Next, we claim that I = J, which is a contradiction. Let g ∈ I. Since IJeX=+1 1 I, for each n ≥ 1,we write g as follows:

αα112 αn 1−1 αn 1 g=+()+ g0 eX1 ⁎⁎ g1 () eX1 g2 +⋯+ () eX1 ⁎ gn−1 + () eX1 ⁎, hn   α1 hgn =+()n eXh11⁎,n+

i αj1 where gg0,,…∈n J and hhnn, +1 ∈ I. Note that if e1 = 0 for some i > 1, then (eX1 ) = 0 for all j ≥ i;so ααi11−1 gg=+()+⋯+0 eXg1 ⁎⁎1 () eX1 gi−1 ∈J. Hence, we assume that e1 is not a nilpotent element. Put r αi1 ﬁ teXgr ≔∑i=0()1 ⁎ i. Then for each N ≥ 1, we can nd a nonnegative integer r such that

αr+1 αr+≤1Γ,Γ, α1 αm ≤ N gt−=r () eX1 11⁎ hr+11 ∈ () eXI ⊆(〚 XE 〛+⋯+〚 X E 〛) for all rN≥−1. Hence by Lemma 2.7, limrr→∞tg= . q We now consider the sequence (gn)n≥0 in J. Since Jf=(1,, … fq), each gk is written as ∑i=1 zfki⁎ i for some Γ,⁎ ≤ zDEki ∈+〚〛. For each r ≥ 0, we obtain

r r  q   r   r  eXαi11⁎⁎⁎⁎⁎⁎⁎. g=()=+⋯+ eXαi z f f z eXαi 1 f z eXαi 1 ∑∑∑∑()1 i  ()1 ij j  1  i11 () q  ∑iq ()1  i====0 i 0  j 1  i 0  i =0 

For each k =…1, , q, put

r ∞ αi αi urk ≔≔∑∑zeXik ⁎and⁎()1 11 uk zeXik ()1 . i==0 i 0

Γ,⁎ ≤ We first show that uk ∈+〚DE〛 for each k =…1, , q. Fix an element kq∈{1, … , }. Note that 0 does not αi appear in supp(zeXik ⁎()1 1 ) for all i ≥ 1, because G()={Γ0}. Let 0supp≠∈βu (k). Note that if β ∈(suppzik ⁎ αi1 fi (eX1 ) ), then β =+iα1 γik for some γzik ∈(supp ik). Therefore, if β belongs to in nitely many supp(zik ⁎ αi1 fi (eX1 ) ), then β ∈⋂i≥11 (iα +Γ), which contradicts Lemma 2.6. Hence, β occurs in only nitely many α i β β zeXik ⁎()1 1 . Thus, for every β ∈(supp uk), the coefficient of X in uk is the sum of coefficients of X in α i only finitely many suitable zeXik ⁎()1 1 . By Lemma 2.9, (Γ, ≤) is artinian; so supp(uk) is artinian. Thus, Γ,⁎ ≤ uk ∈+〚DE〛 for each k =…1, , q. Next, we show that for each k =…1, , q, the sequence (urk) r≥1 converges to uk as r goes to ∞. For each k =…1, , q and each N ≥ 1, there exists a nonnegative integer t such that 1548  Jung Wook Lim and Dong Yeol Oh

αi αΓ,≤≤ α Γ, N uukkt−=∑ zeXXEik ⁎()1 1 ∈(〚〛+⋯+〚〛)1 XEm it≥+1 for all tN≥−1. By Lemma 2.7, urk converges to uk;so

r ∞ αi αi lim∑∑zeXik ⁎()1 11= zeXik ⁎ ()1 . r→∞ i==0 i 0 Therefore,

 r r  gfzeXfzeX=+⋯+lim ⁎() ⁎αj11 ⁎ ⁎ αj  1 ∑∑j11() q ( jq ()1 ) r→∞  j==0 j 0  ∞ ∞ αj αj =+⋯+f1 ⁎⁎∑∑() zj11() eX11 fq ⁎( zjq ⁎() eX1 ) j=0 j=0 ∈ J,

⁎ which proves the claim. Thus, DE+〚 Γ,≤〛 is a Noetherian ring. □

By applying Theorem 2.10 to the case of D = E, we have the following which is the same as [2, 5.5] under the additional condition that (Γ, ≤) is positively ordered.

Corollary 2.11. (cf. [2, 5.5]) Let Γ be cancellative and torsion-free and (Γ, ≤) be a positively narrow ordered monoid. If D is a Noetherian ring and Γ is ﬁnitely generated, then []DΓ,≤ is a Noetherian ring.

Proof. By Lemma 2.4, there exists a compatible strict total order ≤′ on Γ, which is ﬁner than ≤. Note that 0 ≤′ α for all α ∈ Γ. Since ≤′ is ﬁner than ≤ and (Γ, ≤) is narrow,〚DDΓ,≤〛=〚 Γ,≤′〛. By applying Theorem 2.10 to the case of D = E, 〚〛DΓ,≤ is a Noetherian ring. □

Recall that for the usual ordered monoid ,if(Γ,=≤n ) with the lexicographic order, then the ring Γ,⁎ ≤ DE+〚〛 is isomorphic to DX+(11,, … XEXXnn ) [ ,, … ]. By applying Theorems 2.1 and 2.10 to the case when (Γ, ≤) is the monoid n with the lexicographic order, we recover

Corollary 2.12. [4, Theorem 4] Let D ⊆ E be an extension of commutative rings with identity. Then D is a

Noetherian ring and E is a ﬁnitely generated D-module if and only if DX+(11,, … XEXXnn ) [ ,, … ] is a Noetherian ring. In particular, D is Noetherian if and only if DX[…1,, Xn] is Noetherian.

3 Composite generalized power series ring of the form D +[[IΓ⁎,≤]]

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 DI+〚 Γ,≤〛 to be Noetherian when (Γ, ≤) is positively totally ordered. We also give some applications of composite generalized power series rings.

Theorem 3.1. Let D be a commutative ring with identity and let I be a nonzero proper ideal of D. Let (Γ, ≤) be a ⁎ positively totally ordered monoid. Then DI+〚 Γ,≤〛 is a Noetherian ring if and only if D is a Noetherian ring, I 2 = I, and Γ is ﬁnitely generated.

⁎ ⁎⁎ Proof. (⇒) Let DI+〚 Γ,≤〛 be a Noetherian ring. Then DD≅( +〚 IΓ,≤≤ 〛)/〚 I Γ, 〛 is a Noetherian ring. Let ⁎ ⁎ 0 ≠∈a I andα ∈ Γ⁎.SinceDI+〚 Γ,≤〛is Noetherian, the ideal(aXαα,, aX2 …) of DI+〚 Γ,≤〛is ﬁnitely generated; (+)mα1 α mα Γ,⁎ ≤ so there exists a positive integer m such that aXaXaX∈(,, … ).Hence,forsomef1,,…∈+〚fDIm 〛, Noetherian properties in composite generalized power series rings  1549

m (+)mα1 iα aXaXf= ∑ ⁎ i . i=1 Comparing the coefficients of X(+)m 1 α from each side of the equality, we conclude that a = ba for some bI∈ . Hence, II⊆ 2, and thus II= 2. Next, we show that Γ is finitely generated by some modification of the proof of Theorem 2.1(2). Suppose ⁎ that Γ is not finitely generated. Then Γ ≠〈αα1,, … n 〉 for any finite subset {αα1,,… n} of Γ . Since D is a Noetherian ring and I 2 = I, I is a principal ideal generated by an idempotent element a [10, Lemma 1]. For each n ≥ 1, set

ααΓ,⁎⁎≤≤ Γ, IaXDIn ≔1 ( +〚〛)+⋯+ aXDIn ( +〚〛).

α Claim: There exists an element αnn+11∈〈…Γ\αα , , 〉 such that aXIn+1 ∉ n.

Proof of Claim. Suppose, by way of contradiction, that there is no such element αn+1. Let β ∈〈…Γ\αα1 , , n 〉. β β If aXI∉ n, then αn+1 ≔ β contradicts our assumption and so the claim holds. Thus, assume aXI∈ n. ⁎ β n αi Γ,≤ Then aXaXf=∑i=1 ⁎ i, for some f1,,…∈+〚fDEn 〛. Hence, we have

 n  n n αiiα β ∈⊆⋃()=⋃(+())suppaXf∑() ⁎i  supp aXf ⁎i αii supp f ; i=1  i==11i so β =+αγi1 1 for some in1 ∈{1, … , } and γf1 ∈(supp i1 ). Note that γα1 ∉〈1,, … αn 〉 because β ∉〈αα1,, … n 〉.If γ aXI1 ∉ n, then setting αn+1 ≔ γ1 again contradicts our assumption because γα1 ∉〈1,, … αn 〉. Thus, assume γ X 1 ∈ In. We proceed by induction. Assume m ≥ 1 and that γβγγ01=…∈,,,m Γ and ii1 …∈{…,1,,m n} have been chosen so that, for each 1 ≤≤j m,

γj γααj ∉<1,, …n > , aXIγ ∈n ,j−1 = αγγijj +jj , ∈ supp ( fi ). γ γ Note that for β =∉〈…γγ00,,, α1 αn 〉 and X 0 ∈ In. Then aXIm ∈ n implies, as in the argument above, that

γαm =+im+1 γm+1, for some inm+1 ∈{1, 2, … , } and some γfααm+1 ∈()〈…suppinm+1 \1 , , 〉. By the contradiction γ hypothesis, aXIm+1 ∈ n. It follows that we obtain, for every m ≥ 1,

βαααγααααγ=ii12 + +⋯+ immm +m =ii 12 + +⋯+ i + i+ 1 + m+1. Note that since (Γ, ≤) is a (positively) totally ordered monoid, Γ is cancellative by Lemma 2.4. Thus, for every m ≥ 1, γαm =+im+1 γm+1. Hence, for every m ≥ 1, γα γ aXXm = im+1 ⁎, aXm+1 ⁎ and so we have an inﬁnite nondecreasing chain of ideals in DI+〚 Γ,≤〛:

γγγΓ,⁎⁎ Γ, Γ,⁎ aXD1 (+〚〛)⊆⋯⊆ I≤≤≤ aXDmm (+〚〛)⊆ I aXD+1 (+〚〛)⊆⋯ I . ⁎ Since DI+〚 Γ,≤〛 is Noetherian, there exists an integer N ≥ 1 such that, for every kN≥ ,

⁎⁎ aXDγγkN(+〚〛)= IΓ,≤≤ aXD (+〚〛) I Γ, . ⁎ γγN 1 N Γ,≤ Therefore, aXaXf+ = ⁎ , for some f ∈+〚DI〛 and thus γγδNN+1 =+, for some δ ∈(supp f ). Since

γαN =+iN+1 γN+1 and Γ is cancellative, αiN+1 +=δ 0. Since (Γ, ≤) is a positive strictly ordered monoid, ⁎ αiN+1 = 0, which contradicts the fact that αi ∈ Γ . This proves the claim. By the claim, we obtain a strictly Γ,⁎ ≤ Γ⁎,≤ infinite chain II12⊊⊊⋯of ideals of DI+〚〛, which is a contradiction to the fact that DI+[ ] is Noetherian. (⇐) Let D be a Noetherian ring, I 2 = I, and Γ be finitely generated. Then I is principal and is generated by an idempotent element a [10, Lemma 1]. Then DI≅⊕J, where JdaddD={ − | ∈ } is an ideal of D. Since D is a Noetherian ring, ID≅/J and JD≅/I are Noetherian rings. Since Γ is finitely generated, 〚〛IΓ,≤ is ⁎ ⁎ Noetherian [2, 5.5]. Note that DI+〚Γ,≤≤ 〛= JI ⊕〚 Γ, 〛. Thus, DI+〚 Γ,≤〛 is a Noetherian ring. □

⁎ Remark 3.2. Note that an integral domain has only two idempotent elements 0 and 1; so if DI+〚 Γ,≤〛 is a Noetherian domain, then we have either I =(0) or I = D by the proof of Theorem 3.1. Thus, if I is a nonzero ⁎ proper ideal of an integral domain D, then DI+〚 Γ,≤〛 is never a Noetherian domain. 1550  Jung Wook Lim and Dong Yeol Oh

By applying Theorem 3.1 to the case when (Γ,=≤n ) with the lexicographic order, we recover.

Corollary 3.3. [5, Proposition 2.4] Let D be a commutative ring with identity and I a nonzero proper ideal of D. 2 Then D is a Noetherian ring and II= if and only if DX+(11,, … XIXXnn )[ ,, … ] is a Noetherian ring.

Recall that Γ is a numerical semigroup if Γ is a subsemigroup of containing 0 such that it generates as a group. It is well known that a numerical semigroup is ﬁnitely generated. We are closing this paper with some applications of composite generalized power series rings.

Example 3.4. (1) Let be the ring of integers and [i] (resp., [ω]) the ring of Gaussian integers (resp., Eisenstein integers). Then [i] and [ω] are finitely generated as -modules; so if Γ is a numerical semigroup, ⁎ ⁎ then by Theorem 2.10, both +〚 []i Γ,≤〛 and +〚 [ω ]Γ,≤〛 are Noetherian rings. Moreover, if Γ =×⋯×(n times), then by Theorem 2.10 (or Corollary 2.12), +(XXiXX11,, …nn ) [][ ,, … ] and +(XXωXX11,, …nn ) [ ][ ,, … ] are Noetherian rings. (2) Let (resp., ) be the field of real numbers (resp., complex numbers). Then is a finitely generated ⁎ -module; so if Γ is a numerical semigroup, then by Theorem 2.10, +〚Γ,≤〛 is a Noetherian ring. Furthermore, if Γ =×⋯×(n times), then by Theorem 2.10 (or Corollary 2.12), +(XX1,, …n ) [XX1,,… n] is a Noetherian ring. (3) Let R be any Noetherian ring and let n be an integer ≥ 2. Let

 aaa⋯ a a   123nn− 1  a 00⋯ 00     0 aa12⋯ ann−− 2 a 1 00a ⋯ 00        00aaa132⋯ nn−−  00a ⋯ 00  n()=R   aaR1,,…∈nn and ()= R   aR∈ .   ⋮⋮⋮ ⋱⋮⋮  ⋮⋮⋮ ⋱⋮ ⋮         000⋯ aa12  000⋯ a 0        000⋯ 0a  000⋯ 0 a1    

Since n(R) is isomorphic to R, n(R) is a Noetherian ring. Also, n(R) is ﬁnitely generated as an n(R)-module. Thus, by Theorem 2.10 (or Corollary 2.12), nnnn()+(RXX11,, … ) ()[… RXX ,, ] is a Noetherian ring.

n (4) Let R, n, and n(R) be as in (3). Note that RY[]/( Y ) is isomorphic to n(R) [11, Section 1];soby(3), n RX+(11,, … XRYYXn )([ ]/( ))[ ,, … Xn] is a Noetherian ring. (5) Let be the ﬁeld of rational numbers. Then is not ﬁnitely generated as a -module; so by Theorem 2.1 (or Corollary 2.12), +[XX ] is not a Noetherian ring. More precisely, we have a strictly ascending chain of ideals in +[XX ]:

 1   1   1   XXX ⊊⊊⊊⋯    .  2   4   8  (6) Let DYYY=[]/(− 2 ) and I the ideal of D generated by 1 − Y . Then D is a Noetherian ring and I is an idempotent ideal of D;soifΓ is a numerical semigroup, then by Theorem 3.1, DI+[Γ⁎,≤] is a Noetherian ring. Moreover, if Γ =×⋯×(n times), then by Theorem 3.1 (or Corollary 3.3), DX+(1,, … Xn )I [XX1,,… n] is a Noetherian ring.

⁎ (7) Let n be an integer ≥ 2. Then n is not idempotent; so by Theorem 3.1, +〚(n )Γ,≤〛 is not a Noetherian ring for any strictly totally ordered monoid Γ with 0 ≤ α for all α ∈ Γ. In fact, for α ∈ Γ⁎, ⁎ we have a strictly ascending chain of ideals in +〚(n )Γ,≤〛: (nXαααααα)⊊( nX,,, nX223 )⊊( nX nX nX )⊊⋯. In particular, +(Xn )[ X] is not a Noetherian ring (by Corollary 3.3). Noetherian properties in composite generalized power series rings  1551

(8) Let D be a Noetherian ring, E a ring extension of D, I an idempotent ideal of D generated by a, and 0 the ⁎ ⁎ monoid of nonnegative rational numbers. Then neither DE+〚 0,≤〛 nor DI+〚 0,≤〛 is a Noetherian ring. ⁎ ⁎ More precisely, we have strictly ascending chains of ideals of DE+〚 0,≤〛 and DI+〚 0,≤〛, respectively:

1 1 1 ()⊊()⊊()⊊⋯XXX2 4 8 ,   1 1 1 ()⊊()⊊()⊊⋯aX2 aX4 aX8 .

∞ Γ,⁎ ≤ Γ,⁎ ≤ Also, if Γ = ⊕n=1 is the weak direct product of , then neither DE+〚〛 nor DI+〚〛 is a Noetherian ring, i.e., if X ={Xnn | ∈}, then neither DE+[XX] nor DI+[XX] is a Noetherian ring.

Acknowledgments: The authors would like to thank the referees for their several valuable comments and suggestions. D. Y. Oh was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07041083).

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