aa r X i v : . [ m a t h . A C ] J u l ALMOST REDUCTION NUMBER OF CANONICAL IDEALS
SHINYA KUMASHIRO
Abstract.
In this paper, we introduce a new invariant of Cohen-Macaulay lo-cal rings in terms of canonical ideals. The invariant measures how close to beGorenstein, and preserved by localizations, dividing non-zerodivisors, and flat lo-cal homomorphisms. Furthermore it builds bridges between almost Gorensteinand nearly Gorenstein in dimension one. We also explore the invariant in numer-ical semigroup rings and rings arising from idealizations. Introduction
The class of Cohen-Macaulay rings is a central subject in commutative rings. Asis well-known, Cohen-Macaulay rings are stratified byregular rings ⇒ complete intersections ⇒ Gorenstein rings ⇒ Cohen-Macaulay ringsin terms of homological algebra. Among them, the notion of Gorenstein rings isdefined by the local finiteness of the self-injective dimension. Gorenstein rings areknown to have interesting properties such as total reflexivity, and they appear withbeautiful symmetry in not only commutative algebra but also combinatorics, alge-braic geometry, invariant theory and so on. On the other hand, there is a large gapin whether the self-injective dimension is finite or not, and many Cohen-Macaulayrings appearing in concrete examples are actually not Gorenstein rings. For instance,although any normal semigroup rings are Cohen-Macaulay, a normal semigroup ringis Gorenstein only the case that its interior coincides with itself after some shift (see[2, Theorem 6.3.5]). Furthermore, if R is a Cohen-Macaulay local ring and M isa maximal Cohen-Macaulay R -module, the idealization R ⋉ M is always Cohen-Macaulay local ring again. However, R ⋉ M is Gorenstein if and only if M is thecanonical module of R ([20]). Therefore, it seems natural to expect a new class ofrings between Gorenstein and Cohen-Macaulay.One of the important results of the problem is about almost Gorenstein rings .The basic papers [1, 11, 12] revealed the properties of the non-Gorenstein almostGorenstein rings such as G-Regularity and the Gorensteinness of the blow-up of localrings at the maximal ideals. Besides the almost Gorenstein theory, the study of non-Gorenstein Cohen-Macaulay rings has been carried out under intense competition.One can also find other stratifications of Cohen-Macaulay rings in [3, 4, 5, 9, 15]. Mathematics Subject Classification.
Primary: 13H10, Secondary: 13B02, 13D40.
Key words and phrases. almost reduction, canonical ideal, Cohen-Macaulay, Gorenstein, traceideal.The author was supported by JSPS KAKENHI Grant Number JP19J10579. hese theories are based on study of canonical ideals, especially in dimension one.But they have not unified yet.The purpose of this paper is to introduce a new invariant in terms of canonicalideals, say canonical reduction number . The invariant measures how close to beGorenstein, and preserved by localizations, dividing non-zerodivisors, and flat localhomomorphisms (Remark 2.7 (b), (c), Theorem 2.8, and Proposition 2.9). Fur-thermore the invariant will build bridges among above theories of non-GorensteinCohen-Macaulay rings (Theorem 3.11).Let us explain how constructed this paper. In Section 2 we first introduce thenotion of almost reduction as a natural generalization of the notion of reduction.We then define the invariant, the canonical reduction number, by the almost re-duction number of canonical ideals. After that, we investigate the basic propertiesof the canonical reduction number, and characterize rings whose canonical reduc-tion number is less than three. In Section 3 we focus on the case of dimensionone. In this case, the Hilbert function of canonical ideals plays an important role tostudy of Cohen-Macaulay rings. From this perspective, we give a characterization ofthe canonical reduction number in terms of the Hilbert function, and build bridgesbetween almost Gorenstein and nearly Gorenstein as follows. Theorem 1.1. (Theorem 3.11)
Let R be a one-dimensional Cohen-Macaulay localring possessing the canonical module. Then R is almost Gorenstein if and only if R is nearly Gorenstein and the canonical reduction number of R is less than three. In Section 4 we study rings whose canonical reduction number is less than three.In particular, we get a characterization of when the canonical reduction number lessthan three in idealization (Theorem 4.4).Let us fix our notation. Throughout this paper, all rings are Noetherian ringswith identity. For a Noetherian ring R , Q( R ) (resp. R ) denotes the total ring offraction of R (resp. the integral closure of R ). For an R -module M , M ∗ denotes the R -dual Hom R ( M, R ). ℓ R ( M ) denotes the length of M .Suppose that ( R, m ) is a Noetherian local ring and I is an m -primary ideal of R . Then ℓ R ( R/I n ) agrees with a polynomial function of degree d = dim R for all n ≫
0. We then write ℓ R ( R/I n +1 ) = e ( I ) (cid:18) n + dd (cid:19) − e ( I ) (cid:18) n + d − d − (cid:19) + · · · +( − d − e d − ( I ) (cid:18) n + 11 (cid:19) +( − d e d ( I ) with some integers e ( I ), . . . , e d ( I ). The integers e ( I ), . . . , e d ( I ) are called the Hilbert coefficients of I . If R is a Cohen-Macaulay local ring, r( R ) denotes theCohen-Macaulay type of R .We say that an R -module I is a fractional ideal , if I is a finitely generated R -submodule of Q( R ) containing a non-zerodivisor of R . For fractional ideals I and J , I : J (resp. I : R J ) stands for the set { α ∈ Q( R ) | αJ ⊆ I } (resp. ( I : J ) ∩ R = { α ∈ R | αJ ⊆ I } ). We freely use the following facts. Remark 1.2. ([14]) Let R be a Noetherian ring. Let I and J be fractional ideals.Then we have the followings. a) If I ∼ = J , then J = αI for some α ∈ Q( R ).(b) I : J ∼ = Hom R ( J, I ), where α ∈ I : J corresponds to the multiplication map by α . 2. almost reduction and canonical reduction number Let R be a Noetherian ring. First of all, we introduce the notion of almostreduction. Definition 2.1.
Let I and J be ideals of R . Then we say that J is an almostreduction of I if there exists an integer ℓ ≥ J ℓ ⊆ I ℓ and(b) I ℓ +1 = J I ℓ .Almost reduction has the same properties as reduction. Remark 2.2.
Let I and J be ideals of R . Suppose that J is an almost reductionof I . Then we have the followings.(a) √ I = √ J , where √ I denotes the radical of I . Hence height R I = height R J .(b) If J ℓ ⊆ I ℓ and I ℓ +1 = J I ℓ for ℓ ≥
0, then J n ⊆ I n and I n +1 = J I n for all n ≥ ℓ .(c) Let ( R, m ) be a Noetherian local ring. If I and J are m -primary ideals of R ,then e ( J ) = e ( I ). Proof. (a): It follows from the definition that J ℓ ⊆ I ℓ and I ℓ +1 = J I ℓ ⊆ J for some ℓ ≥ n ≥ ℓ , we have I n +1 = I n − ℓ · I ℓ +1 = I n − ℓ · J I ℓ = J I n . We then have I n = J n − ℓ · I ℓ ⊇ J n − ℓ · J ℓ recursively.(c): Choose ℓ ≥ J ℓ ⊆ I ℓ and I ℓ +1 = J I ℓ . Then, for all n >
0, we have J ℓ + n ⊆ I ℓ + n = J n I ℓ ⊆ J n . It follows that ℓ R ( R/J n ) ≤ ℓ R ( R/I ℓ + n ) ≤ ℓ R ( R/J ℓ + n ). Hence we have e ( J ) = e ( I )by diverging n → ∞ . (cid:3) It is a well-known fact that, for a Noetherian local ring ( R, m ) with the infinitefield R/ m , each m -primary ideal I has a parameter ideal Q ⊆ I as its reduction.However, it is not true if R/ m is finite. The following is an example of an m -primaryideal which has no parameter reduction, but has a parameter almost reduction. Example 2.3. (cf. [11, Remark 2.10]) Let k [[ X, Y, Z ]] be the formal power seriesring over the field k = Z / Z . Set R = k [[ X, Y, Z ]] / a , where a = ( X, Y ) ∩ ( Y, Z ) ∩ ( Z, X ) = (
XY, Y Z, ZX ) . Set I = ( x + y, y + z ), where x , y , and z denote the images of X , Y , and Z in R respectively. Then ( x + y + z ) is an almost reduction of I , but I has no parameterreduction. Proof.
Set m = ( x, y, z ) and f = x + y + z . Then I = m = I + ( f ), whence f I . Astandard computation shows I n = ( x n , y n , z n ) for all n ≥
2. Hence we have f I = I and f ∈ I , which follows that f is an almost reduction of I . Assume that ( a ) is reduction of I . Write a = c x + c y + c z + g , where c , c , c ∈ k = Z / Z and g ∈ m . Then we obtain aI n = ( c x n +1 , c y n +1 , c z n +1 ) = I n +1 for n ≫
0. Therefore, c = c = c = 1. It follows that f ∈ I since a = f + g ∈ I ,thus it is a contradiction. (cid:3) In Section 3 we explore the existence of almost reduction for m -primary idealsover one-dimensional Cohen-Macaulay local rings. In what follows, let us focus onthe case where the height of ideal is one. Proposition 2.4.
Let R be a Noetherian ring. Let I and J be ideals of R containinga non-zerodivisor of R . Suppose that ( a ) and ( b ) are almost reductions of I and J respectively. If I ∼ = J , then we have the followings. (a) For any ℓ ≥ , ( a ) ℓ ⊆ I ℓ and I ℓ +1 = aI ℓ if and only if R [ Ia ] = ( Ia ) ℓ . (b) R [ Ia ] = R [ Jb ] in Q( R ) . (c) For any ℓ ≥ , ( a ) ℓ ⊆ I ℓ and I ℓ +1 = aI ℓ if and only if ( b ) ℓ ⊆ J ℓ and J ℓ +1 = bJ ℓ .Proof. Note that a is a non-zerodivisor of R since I n +1 = aI n ⊆ ( a ) for n ≫ L = Ia and L = Jb . Then R ⊆ L n = L n +11 and R ⊆ L n = L n +12 for n ≫ ≤ t ≤ n −
1, we have R · L t ⊆ L n + t = L n . Hence R [ L ] = L n and R [ L ] = L n .By Remark 1.2 (1), we have L = αL for some α ∈ Q( R ). It follows that R [ L ] = αR [ L ] by substituting L = αL to L n = L n +12 .(a): The above argument actually shows that the “only if” part. Conversely,suppose that R [ L ] = L ℓ . If ℓ = 0, then I ⊆ ( a ). Hence I = aI for some ideal I .Then ( a ) n ⊆ I n = a n I n ⊆ ( a ) n for n ≫
0, whence I = R . Thus I = ( a ). Assume ℓ >
0. Then L ℓ +11 and L ℓ − are in R [ L ] = L ℓ . The latter implies that L ℓ ⊆ L ℓ +11 ,thus L ℓ = L ℓ +11 . It follows that I ℓ +1 = aI ℓ . Furthermore we have ( a ) ℓ ⊆ I ℓ since R ⊆ R [ L ] = L ℓ .(b): It follows from the observation that R [ L ] = L n = α n L n = α n R [ L ] = R [ L ].(c): We have only to show the “only if” part. By the assumption and (a) we have L ℓ = R [ L ]. Hence L ℓ = α ℓ L ℓ = α ℓ R [ L ] = R [ L ] = R [ L ]by (b). It follows that ( b ) ℓ ⊆ J ℓ and J ℓ +1 = bJ ℓ by (a). (cid:3) Proposition 2.4 (c) claims that the almost reduction number of I and that of J are equal. This fact provides a new invariant of Cohen-Macaulay local rings. Let usrecall the following fact. Fact 2.5. ([2, Proposition 3.3.18]) Let ( R, m ) be a Cohen-Macaulay local ring pos-sessing the canonical module ω R . Then the following conditions are equivalent:(a) R is generically Gorenstein, that is, R p is Gorenstein for all p ∈ Min R ;(b) ω R has a rank;(c) there exists an ideal ω ⊆ R such that ω ∼ = ω R .When this is the case, if ω ( R , then height R ω = 1 and R/ω is Gorenstein. e call an ideal ω ⊆ R is canonical if ω is isomorphic to the canonical module of R . Definition 2.6.
Let ( R, m ) be a Cohen-Macaulay local ring possessing the canonicalmodule. Suppose that R is generically Gorenstein. We then callinf (cid:26) n ≥ (cid:12)(cid:12)(cid:12)(cid:12) there exist a canonical ideal ω and an almost reduction ( a ) of ω such that ω n +1 = aω n and ( a ) n ⊆ ω n (cid:27) the canonical reduction number of R , and denote by can . red R . Remark 2.7.
Let ( R, m ) be a Cohen-Macaulay local ring possessing the canonicalmodule. Suppose that there exist a canonical ideal ω and an almost reduction ( a )of ω . Then we have the followings.(a) ω n +1 = aω n implies that ( a ) n ⊆ ω n .(b) can . red R p ≤ can . red R for all p ∈ Spec R .(c) Suppose that x ∈ m is a non-zerodivisor of R and R/ω . Then can . red R/xR ≤ can . red R . Proof. (a) follows from (cid:0) ωa (cid:1) n = (cid:0) ωa (cid:1) m = R [ ωa ] ⊇ R for m ≫ n by Proposition 2.4 (a).(b) follows from the fact that ωR p is a canonical ideal of R p . (c) follows from the factthat ( ω + ( x )) / ( x ) is a canonical ideal of R/ ( x ) since ( ω + ( x )) / ( x ) ∼ = ω/ (( x ) ∩ ω ) = ω/xω . (cid:3) Furthermore flat base change of rings preserves the canonical reduction number.
Theorem 2.8.
Let ( R, m ) → ( S, n ) be a flat local homomorphism of Cohen-Macaulayrings such that S/ m S is Gorenstein. Suppose that there exists the canonical mod-ule ω R of R and R is generically Gorenstein. If can . red R < ∞ , then can . red R =can . red S .Proof. Note that we have ω S ∼ = S ⊗ R ω R since S/ m S is Gorenstein. Hence, if ω is acanonical ideal of R , then ωS is a canonical ideal of S .Suppose that can . red R < ∞ . Then there exist a canonical ideal ω and its almostreduction ( a ). Let n ≥
0. Then we obtain that n ≥ can . red R ⇔ R (cid:2) ωa (cid:3) / (cid:0) ωa (cid:1) n = 0 ⇔ S ⊗ R (cid:0) R (cid:2) ωa (cid:3) / (cid:0) ωa (cid:1) n (cid:1) = 0 ⇔ S (cid:2) ωSa (cid:3) / (cid:0) ωSa (cid:1) n = 0 ⇔ n ≥ can . red S by Proposition 2.4 (a). Therefore, we have can . red R = can . red S . (cid:3) The canonical reduction number measures how close to be Gorenstein. In whatfollows, unless otherwise stated, let ( R, m ) be a Cohen-Macaulay local ring possessingthe canonical module ω R . Set d = dim R . Proposition 2.9.
The following conditions are equivalent: (a) R is Gorenstein; (b) can . red R = 0 ; (c) can . red R ≤ ; (d) there exist a canonical ideal ω ⊆ R and a ∈ R such that ω = aω . roof. Since (a) ⇒ (b) ⇒ (c) ⇒ (d) is trivial, we have only to show that (d) ⇒ (a).Note that ω = aω implies that ( ωa ) n = ωa for all n >
0. Hence R [ ωa ] = R + ωa . Itfollows that R [ ωa ] = R since R (cid:2) ωa (cid:3) ⊆ ωa : ωa = R. Thus ωa ⊆ R . Therefore, we get ω = aω for some canonical ideal ω ⊆ R . Thisshows that ω = ω by ω = aω . Since ω is nonzero, it forces that ω = R , whence R is Gorenstein. (cid:3) Corollary 2.10.
Suppose that R is a Cohen-Macaulay local normal domain. Then can . red R < ∞ if and only if R is Gorenstein.Proof. We have only to show that the only if part. If can . red R < ∞ , then wecan choose a canonical ideal ω and its reduction ( a ). By Proposition 2.4, we have R ⊆ R [ ωa ] ⊆ R , where R denotes the integral closure of R . Thus can . red R = 0. (cid:3) We next characterize a ring R with can . red R ≤
2. To state our theorem, let usrecall the definition of trace ideals.
Definition 2.11.
For an R -module M , the image of the evaluation map ϕ : Hom R ( M, R ) ⊗ R M → R, where ϕ ( f ⊗ x ) = f ( x ) for f ∈ Hom R ( M, R ) and x ∈ M , is called the trace ideal of M and denoted by tr R ( M ). We say that an ideal I is a trace ideal of R if I = tr R ( M ) for some R -module M . Remark 2.12. (a) If I is a fractional ideal, then tr R ( I ) = ( R : I ) I .(b) ([7, Corollary 2.2]) Let R be a Noetherian ring and I an ideal containing anon-zerodivisor of R . Then the following conditions are equivalent:(i) I is a trace ideal of R ;(ii) I = ( R : I ) I , that is, I = tr R ( I );(iii) I : I = R : I .(c) ([15, Lemma 2.1]) The trace ideal tr R ( ω R ) of the canonical module describes thenon-Gorenstein locus of R , that is, { p ∈ Spec R | R p is not Gorenstein } = { p ∈ Spec R | tr R ( ω R ) ⊆ p } . Proof. (a): It follows from the fact that the evaluation map Hom R ( I, R ) ⊗ R I → R isidentified with the map ( R : I ) ⊗ R I → R , where f ⊗ x f x by Remark 1.2. (cid:3) The following is a characterization of the canonical reduction number two.
Theorem 2.13.
The following conditions are equivalent: (a) can . red R ≤ ; (b) tr R ( ω R ) ∼ = ω ∗ R . To prove Theorem 2.13, we prepare a lemma.
Lemma 2.14. If d = 0 , then tr R ( ω R ) ∼ = ω ∗ R if and only if R is Gorenstein. roof. If R is Gorenstein, then tr R ( ω R ) = R ∼ = ω ∗ R by Remark 2.12. Conversely, sup-pose tr R ( ω R ) ∼ = ω ∗ R . Note that ω ∗ R ∼ = Hom R ( ω R , Hom R ( ω R , ω R )) ∼ = Hom R ( ω R ⊗ R ω R , ω R ).Hence, by applying the ω R -dual to tr R ( ω R ) ∼ = ω ∗ R , we haveHom R (tr R ( ω R ) , ω R ) ∼ = ω R ⊗ R ω R . (1)On the other hand, by applying the ω R -dual to the exact sequence 0 → tr R ( ω R ) → R → R/ tr R ( ω R ) →
0, we have a surjection ω R → Hom R (tr R ( ω R ) , ω R ) . (2)Therefore, from (1) and (2), we obtain the surjection ω R → ω R ⊗ R ω R . It followsthat r( R ) ≥ r( R ) , whence R is Gorenstein. (cid:3) Corollary 2.15. If tr R ( ω R ) ∼ = ω ∗ R , then R is generically Gorenstein.Proof. It is a direct consequence of Lemma 2.14 and [16, Proposition 2.8 (viii)]. (cid:3)
Proof of Theorem 2.13.
Note that we may assume that d > ⇒ (b): Choose a canonical ideal ω ( R and a ∈ R so that ω = aω and a ∈ ω . Set K = ωa . Then we have R [ K ] = K by Proposition 2.4 (a). We obtainthat R : R [ K ] = R : K = ( K : K ) : K = K : K = K : K = ( K : K ) : K = R : K. It follows that( R : K ) K = ( R : R [ K ]) K ⊆ ( R : R [ K ]) R [ K ] = R : R [ K ] = R : K. Furthermore we have ( R : K ) K ⊆ ( R : K ) K , whence R : K ⊆ ( R : K ) K since ( R : K ) K = ( R : R [ K ]) R [ K ] = R : R [ K ] = R : K . Therefore, we gettr R ( ω R ) = ( R : K ) K = R : K ∼ = ω ∗ R by Remark 2.12(a).(b) ⇒ (a): Due to Corollary 2.15, we may assume that there exists a canonicalideal ω ( R . Then we have tr R ( ω R ) = ( R : ω ) ω and ω R ∼ = R : ω . Hence there exists α ∈ Q( R ) such that ( R : ω ) ω = α ( R : ω ). It follows that ( R : ω ) ω n = α n ( R : ω ) forall n >
0. We then obtain that (cid:0) ωα (cid:1) n ⊆ ( R : ω ) : ( R : ω ) = R : ( R : ω ) ω = ( ω : ω ) : α ( R : ω )= ω : α ( R : ω ) ω = ω : α ( R : ω ) = ω : α ( ω : ω )= α ( ω : ( ω : ω )) = (cid:0) ωα (cid:1) for all n > n = 1, we have ωα ⊆ (cid:0) ωα (cid:1) , whence αω ⊆ ω . By substituting n = 3, we have (cid:0) ωα (cid:1) ⊆ (cid:0) ωα (cid:1) , whence ω ⊆ αω . Hence ω = αω . Furthermore, bysubstituting n = 2, we have 1 ∈ ( R : ω ) : ( R : ω ) = (cid:0) ωα (cid:1) . Hence α ∈ ω .The rest of the proof is to replace α to an element in R . Write α = ab for somenon zerodivisors a, b of R . Then ω = bω ( R is also a canonical ideal of R , and wehave ω = aω and a ∈ ω as desired. (cid:3) . One-dimensional case
Let ( R, m ) be a Cohen-Macaulay local ring of dimension one. Suppose that R possesses the canonical module ω R and a canonical ideal ω ( R . Then, sincedim R = 1, ω is an m -primary ideal of R . Hence we can define the Hilbert function ℓ R ( R/ω n ) of canonical ideal ω . In [3, 11], the authors have explored the notions ofalmost Gorenstein and 2-almost Gorenstein by the analysis of the Hilbert function ℓ R ( R/ω n ). With this background, this section focuses on the case of dimension one.First of all, we establish the existence of almost reduction of a canonical ideal(Corollary 3.5). Definition 3.1. ([18, Proposition 1.1] and [19, before Lemma 8.2]) Let R be aNoetherian ring and I an ideal of R . We then set(a) e I = S ℓ> ( I ℓ +1 : R I ℓ ) and(b) R I = S ℓ> ( I ℓ : I ℓ ).The ideal e I is called the Ratliff-Rush closure of I , and the ring R I is coinsides withthe blow-up of R at I when dim R = 1. Lemma 3.2. ([19, Lemma 8.2])
Let R be a Noetherian ring and I an ideal of R containing a non-zerodivisor of R . Then I n = ( e I ) n = e I n for all n ≫ . Lemma 3.3. (cf. [18, Proposition 1.1])
Let ( R, m ) be a Cohen-Macaulay local ringof dimension one and I an m -primary ideal of R . Then we have the followings. (a) If there exists a reduction ( a ) ⊆ I of I , then R I = R [ Ia ] = (cid:0) Ia (cid:1) n for all n ≫ . (b) R I = I n : I n for all n ≫ . (c) R I = R I n for all n > . (d) IR I ∼ = R I . Proposition 3.4.
Let ( R, m ) be a Cohen-Macaulay local ring of dimension one and I an m -primary ideal of R . Then there exist an m -primary ideal J and a ∈ R suchthat J ∼ = I and ( a ) is an almost reduction of J .Proof. Since Lemma 3.3 (b) and (c), we have R I = R I n = I n : I n for all n ≫ I n R I n = ( I n : I n ) I n = I n is isomorphic to R I n = R I by Lemma 3.3 (d). Itfollows that I n +1 ∼ = IR I ∼ = R I ∼ = I n . Hence I n +1 = αI n for some α ∈ Q( R ). Write α = ab for non-zerodivisors a and b in R . Then we obtain ( bI ) n +1 = a ( bI ) n and a ∈ ( bI ) n +1 : R ( bI ) n ⊆ e bI . Therefore, wehave a n ∈ ( e bI ) n = ( bI ) n for n ≫ a ) is an almost reductionof bI . (cid:3) Due to Proposition 3.4 we have the following.
Corollary 3.5.
Let R be a one-dimensional Cohen-Macaulay local ring. If R has acanonical ideal, then can . red R < ∞ . While Proposition 3.4 holds, there exists an m -primary ideal which has no param-eter almost reduction. xample 3.6. ([18, after Definition 2.1]) Let R = k [[ X, Y ]] / ( XY ( X + Y )), where k = Z / Z . Let x, y denote the images of X, Y in R . Then any element of R is notan almost reduction of m = ( x, y ). Proof.
Suppose that m has an almost reduction ( a ). Then, since a n ∈ m n ⊆ m for n ≫ a ∈ m . It follows that ( a ) is a reduction of m . Write a = c x + c y + g , where c , c ∈ k and g ∈ m . Then we can replace g by 0. Actually, we have a m n ⊆ ( a, g ) m n = ( a − g ) m n + g m n ⊆ ( a − g ) m n + m n +2 ⊆ m n +1 and thus ( a − g ) m n = m n +1 for n ≫ a is either x , y , or x + y . It concludes that a is a zerodivisor of R , which is acontradiction since m n +1 ⊆ ( a ). (cid:3) Let us continue to explore the canonical reduction number in dimension one.
Proposition 3.7.
Let ( R, m ) be a Cohen-Macaulay local ring of dimension one.Suppose that R possesses a canonical ideal ω ( R . We then have the followings. (a) The following integers are equal: (i) can . red R ; (ii) min { m ≥ | ℓ R ( R/ω n ) agrees with the polynomial function for all n ≥ m } .Furthermore, if R/ m is infinite, then the above two integers also equal to (iii) min (cid:26) n ≥ (cid:12)(cid:12)(cid:12)(cid:12) there exist a canonical ideal ω and a reduction ( a ) of ω such that ω n +1 = aω n (cid:27) . (b) 0 ≤ can . red R ≤ e ( m ) − , where e ( m ) denotes the multiplicity of R .Proof. (a): The equality of (i) and (iii) follows from Proposition 2.4 (c). To showthe equality of (i) and (ii) we may assume that R is not Gorenstein by Proposition2.9. Choose a canonical ideal ω ( R so that ω has an almost reduction ( a ). Set c = can . red R >
0. Then, for all n ≥ c , we have ℓ R ( R/ω n ) = ℓ R ( R/ ( a ) n ) − ℓ R ( ω n / ( a ) n )= ℓ R ( R/ ( a )) · n − ℓ R (cid:0)(cid:0) ωa (cid:1) n /R (cid:1) = e ( ω ) n − ℓ R (cid:0) R (cid:2) ωa (cid:3) /R (cid:1) by Remark 2.2 (c) and Proposition 2.4 (a). Note that R (cid:2) ωa (cid:3) is independent of thechoice of almost reductions by Proposition 2.4 (b). Thus c ≥ the integer of (ii).Assume that c > the integer of (ii). Then ℓ R ( R/ω c − ) = ℓ R ( R/ω c ) − ℓ R ( ω c − /ω c )= e ( ω ) c − ℓ R (cid:0) R (cid:2) ωa (cid:3) /R (cid:1) − ℓ R (cid:16)(cid:0) ωa (cid:1) c − /aR (cid:2) ωa (cid:3)(cid:17) > e ( ω ) c − ℓ R (cid:0) R (cid:2) ωa (cid:3) /R (cid:1) − ℓ R (cid:0) R (cid:2) ωa (cid:3) /aR (cid:2) ωa (cid:3)(cid:1) . On the other hand, we have ℓ R ( R/ω c − ) = e ( ω )( c − − ℓ R (cid:0) R (cid:2) ωa (cid:3) /R (cid:1) since c > the integer of (ii). It follows thate ( ω ) < ℓ R (cid:0) R (cid:2) ωa (cid:3) /aR (cid:2) ωa (cid:3)(cid:1) = e ( ω ) · rank R R (cid:2) ωa (cid:3) = e ( ω )by the multiplicative formula. This is a contradiction. b): To prove the inequality we may assume that R/ m is infinite by Theorem 2.8.Then it follows from (a) and [6]. (cid:3) Next we study a relation between almost Gorenstein rings and nearly Gorensteinrings in terms of the canonical reduction number. In what follows, throughout thissection, let ( R, m ) be a one-dimensional Cohen-Macaulay local ring with a canonicalideal. Choose a canonical ideal ω ( R so that ω has an almost reduction ( a ). Westart to recall the definitions of almost Gorenstein and nearly Gorenstein. Definition 3.8. (a) ([11, Definition 3.1]) We say that R is almost Gorenstein ife ( ω ) ≤ r( R ).(b) ([15, Definition 2.2]) We say that R is nearly Gorenstein if tr R ( ω R ) ⊇ m .We reconstruct a characterization of almost Gorenstein rings by [1]. Note that[11, Theorem 3.11] assumes that there exists a parameter reduction of a canonicalideal, see [11, Setting 3.4]. Proposition 3.9. (cf. [11, Theorem 3.11])
The following conditions are equivalent: (a) R is almost Gorenstein; (b) m R [ ωa ] = m ; (c) m ω ⊆ ( a ) .Proof. We may assume that R is not Gorenstein. Set K = ωa .(b) ⇔ (c): It follows from the equivalences m ω ⊆ ( a ) ⇔ m K ⊆ R ⇔ m K ( R ⇔ m K ⊆ m ⇔ m K n ⊆ m for all n > ⇔ m R [ K ] ⊆ m ⇔ m R [ K ] = m , where the second equivalence follows from the fact that m K = R implies m isprincipal, that is, R is a discrete valuation ring.(a) ⇔ (b): Note that R [ K ] is independent of the choice of ω and ( a ) by Proposition2.4 (b). Hence, after enlarging the residue field R/ m , we may assume that R/ m isinfinite and ( a ) is a reduction of ω . Therefore, we have the conclusion since R is almost Gorenstein ⇔ m K ⊆ R ⇔ m R [ K ] = m , where the first equivalence follows from [11, Theorem 3.11] and the second equiva-lence follows from the proof of (b) ⇔ (c). (cid:3) Lemma 3.10. If R/ ( R : R [ ωa ]) is Gorenstein, then can . red R ≤ .Proof. Set K = ωa and S = R [ K ]. By applying the K -dual K : − to the short exactsequence 0 → R : S → R → R/ ( R : S ) →
0, we have0 → K → K : ( R : S ) → Ext R ( R/ ( R : S ) , K ) → . On the other hand, we obtain K : ( R : S ) = K : (( K : K ) : S ) = K : ( K : KS ) = K : ( K : S ) = S. Hence ω R/ ( R : S ) ∼ = Ext R ( R/ ( R : S ) , K ) ∼ = S/K . By our assumption we have S = K + Rs for some s ∈ S . Let α ∈ K . Then, since sK ⊆ S = K + Rs , here exists β ∈ K and r ∈ R such that sα = β + rs . Whence s ( α − r ) = β ∈ K .It follows that( α − r ) K ⊆ ( α − r ) S = ( α − r )( K + Rs ) = ( α − r ) K + Rβ ⊆ K . Therefore, for α ∈ K and α ′ ∈ K , there are elements r ∈ R and α ′′ ∈ K such that( α − r ) α ′ = α ′′ . Thus αα ′ = rα ′ + α ′′ ∈ K . It concludes that K = K . Therefore,we have K = K n = R [ K ] for n ≫
0, that is, can . red R ≤ (cid:3) Now we can illustrate a relation between almost Gorenstein and nearly Gorenstein(see [15, Theorem 7.4]).
Theorem 3.11.
The following conditions are equivalent: (a) R is almost Gorenstein; (b) R is nearly Gorenstein and can . red R ≤ ; (c) R is nearly Gorenstein and R/ ( R : R [ ωa ]) is Gorenstein.Proof. For each proof of implication, we may assume that R is not Gorenstein. Set K = ωa and S = R [ K ].(a) ⇒ (c): By Proposition 3.9, we have m S = m ⊆ R . It follows that m ⊆ R : S ⊆ R : K ⊆ ( R : K ) K = tr R ( ω R ) ( R . Hence R is nearly Gorenstein and R/ ( R : S ) = R/ m is Gorenstein.(c) ⇒ (b): It follows from Lemma 3.10.(b) ⇒ (a): By Proposition 2.4 (a), K = S . It follows that R : K = ( K : K ) : K = K : K = K : S = K : KS = ( K : K ) : S = R : S . Hence m = tr R ( ω R ) = ( R : K ) K = ( R : S ) K = R : S, where the fourth equivalence follows from ( R : S ) K ⊆ ( R : S ) S = R : S and R : S = ( R : S ) K ⊆ ( R : S ) K . Therefore, we get m K ⊆ m S ⊆ R . (cid:3) Note that the ring in Example 2.3 is almost Gorenstein which has no parameterreduction ([11, Example 3.2 (1)]). Thus Theorem 3.11 is stated in pretty generalsetting than [15, Theorem 7.4].In the rest of this section we note an example arising from numerical semigrouprings. Let us recall several notations of numerical semigroup rings.
Definition 3.12.
Let 0 < a , a , . . . , a ℓ ∈ Z ( ℓ >
0) be positive integers withGCD ( a , a , . . . , a ℓ ) = 1. Set H = h a , a , . . . , a ℓ i = ( ℓ X i =1 c i a i | ≤ c i ∈ Z for all 1 ≤ i ≤ ℓ ) and call it the numerical semigroup generated by a , . . . , a ℓ . Let k [[ t ]] be the formalpower series ring over a field k . We then set R = k [[ H ]] = k [[ t a , t a , . . . , t a ℓ ]]in k [[ t ]] and call it the semigroup ring of H over k . The ring R is a one-dimensionalCohen-Macaulay local domain with m = ( t a , t a , . . . , t a ℓ ). et PF( H ) = { n ∈ Z \ H | n + a i ∈ H for all 1 ≤ i ≤ ℓ } denote the set of pseudo-Frobenius numbers of H . Write PF( H ) = { c < c < · · · Let n ≥ 3, and set H = h n, n + 1 , n − n − i . Then PF( H ) = { n − n − , n − n − } . Hence the fractional canonical ideal of R = k [[ H ]] is K = R + Rt . Therefore, we get can . red R = n − properties of a ring R with tr R ( ω R ) ∼ = ω ∗ R Let ( R, m ) be a Cohen-Macaulay local ring possessing the canonical module ω R .In this section we investigate the condition tr R ( ω R ) ∼ = ω ∗ R . Due to Theorem 2.13,we have the following. Proposition 4.1. Let ( R, m ) be a Cohen-Macaulay local ring possessing the canon-ical module. Suppose that there exist a canonical ideal ω and its almost reduction.Then we have the followings. (a) tr R p ( ω R p ) ∼ = ω ∗ R p for all p ∈ Spec R . (b) Suppose that x ∈ m is a non-zerodivisor of R and R/ω . Then tr R/ ( x ) ( ω R/ ( x ) ) ∼ = ω ∗ R/ ( x ) . (c) Let ( R, m ) → ( S, n ) be a flat local homomorphism of Cohen-Macaulay rings suchthat S/ m S is Gorenstein. Suppose that there exists the canonical module ω R of R . Then tr R ( ω R ) ∼ = ω ∗ R if and only if tr S ( ω S ) ∼ = ω ∗ S .Proof. It follows from Remark 2.7 (b), (c), Theorem 2.8, and Theorem 2.13. (cid:3) Remark 4.2. Note that Proposition 4.1 (b) and the if part of (c) may not followimmediately from general properties of trace ideals.For a moment, let R be an arbitrary commutative ring and M an R -module. Let A = R ⋉ M denote the idealization of M over R , that is, A = R ⊕ M as an R -moduleand the multiplication in A is given by( a, x )( b, y ) = ( ab, bx + ay )where a, b ∈ R and x, y ∈ M . The followings are fundamental. Fact 4.3. For a local ring R and a nonzero R -module M , we have the followings: a) R ⋉ M is a Noetherian ring if and only if R is a Noetherian ring and M is afinitely generated R -module.(b) R ⋉ M is a Cohen-Macaulay ring if and only if R is a Cohen-Macaulay ring and M is a maximal Cohen-Macaulay R -module.(c) ([20, (7) Theorem]) R ⋉ M is a Gorenstein ring if and only if R is a Cohen-Macaulay ring possessing the canonical module ω R and M ∼ = ω R .The Fact 4.3 shows that the ring structures of R ⋉ M corresponds to the propertiesof R and M . Especially, if R is a Cohen-Macaulay local ring and M is a maximalCohen-Macaulay R -module, the idealization R ⋉ M builds bridges between the strat-ification of Cohen-Macaulay rings and the classification of maximal Cohen-Macaulaymodules.Set A = R ⋉ M . The following theorem gives a characterization of when tr A ( ω A ) ∼ = ω ∗ A via the properties of M . The theorem also shows that the class of the ring R with tr R ( ω R ) ∼ = ω ∗ R is abundant. Recall that B is a finite birational extension of R if B is a subring of Q( R ) containing R and finitely generated as an R -module. Theorem 4.4. Let ( R, m ) be a Gorenstein local ring of dimension d > . Let M bea maximal Cohen-Macaulay faithful R -module. Set A = R ⋉ M . Then the followingassertions are equivalent: (a) tr A ( ω A ) ∼ = ω ∗ A ; (b) can . red A ≤ ; (c) M is isomorphic to some trace ideal of R ; (d) M ∗ is isomorphic to some finite birational extension B of R such that B M is aCohen-Macaulay local ring of dimension d for all M ∈ Max B .When this is the case, if A is not Gorenstein, then r( A ) = r( R/I ) + 2 where I denotes the trace ideal isomorphic to M . To prove Theorem 4.4, we prepare several propositions. Lemma 4.5. Let ( R, m ) be a Noetherian local ring of dimension d . Let B be a finitebirational extension of R . Then the following conditions are equivalent: (a) B is a maximal Cohen-Macaulay R -module; (b) B M is a Cohen-Macaulay local ring of dimension d for all M ∈ Max B .Proof. It is known for professional, but we include the proof for the convenience ofreader.(a) ⇒ (b): Let M ∈ Max B . Note that depth B M ≥ depth R B since a B -regularsequence in m is a B M -regular sequence in M . It follows that d = dim B ≥ dim B M ≥ depth B M ≥ depth R B = d, thus B M is a Cohen-Macaulay local ring of dimension d .(b) ⇒ (a): Note that B is a semilocal ring since B is a finite birational extensionof R . Hence m B ⊆ J ( B ) ⊆ √ m B , where J ( B ) denotes the Jacobson radical of B and √ m B denotes the radical of m B . Hence grade( m B, B ) = grade( J ( B ) , B ) = d .It follows that there exists a B -regular sequence in m of length d . (cid:3) roposition 4.6. (cf. [7, Corollary 2.8]) Let ( R, m ) be a Gorenstein local ring ofdimension d > . Set ϕ : (cid:26) I (cid:12)(cid:12)(cid:12)(cid:12) I is a trace ideal containing a non-zerodivisor of R and maximal Cohen-Macaulay as an R -module (cid:27) → (cid:26) B (cid:12)(cid:12)(cid:12)(cid:12) B is a finite birational extension of R such that B M isa Cohen-Macaulay local ring of dimension d for all M ∈ Max B (cid:27) , where I I : I . Then ϕ is a one-to-one correspondence.Proof. (well-definedness): By Remark 2.12 (b), we have I : I = R : I ∼ = I ∗ . Hence I : I is a maximal Cohen-Macaulay R -module since R is Gorenstein. Hence ( I : I ) M is a Cohen-Macaulay ring of dimension d for all M ∈ Max( I : I ) by Lemma 4.5.(injective): Let I and J be trace ideals containing non-zerodivisors of R andmaximal Cohen-Macaulay R -modules. If I : I = J : J , then we have I = R : ( R : I ) = R : ( I : I ) = R : ( J : J ) = R : ( R : J ) = J .(surjective): Let B be a finite birational extension of R such that B M is a Cohen-Macaulay local ring of dimension d for all M ∈ Max B . Then, by Lemma 4.5, B isa maximal Cohen-Macaulay R -module. Hence so is R : B . R : B is a trace idealsince tr R ( B ) = ( R : B ) B = R : B . Furthermore ( R : B ) : ( R : B ) = R : ( R : B ) B = R : ( R : B ) = B as desired. (cid:3) Proposition 4.7. Let ( R, m ) be a Gorenstein local ring of dimension d > and I atrace ideal of R . Suppose that I is maximal Cohen-Macaulay as an R -module. Thenthere exists an element x ∈ m such that x is a non-zerodivisor of R and R/I , and I · R/ ( x ) is a trace ideal of R/ ( x ) and maximal Cohen-Macaulay as an R/ ( x ) -module.Proof. By applying the depth formula to the exact sequence0 → I ι −→ R → R/I → , (3)we have depth R R/I ≥ d − > 0. Hence we can choose x ∈ m so that x is anon-zerodivisor of R and R/I .Note that the embedding ι in (3) induces the isomorphismHom R ( I, I ) ∼ = Hom R ( I, R )(4)by [17, Lemma 2.3] (or see [7, Proposition 2.1]). Set ∗ = R/ ( x ) ⊗ R ∗ . The goal isto prove that the map Hom R ( IR, IR ) → Hom R ( IR, R ) induced by ι : IR → R isbijective. Now we have Hom R ( I, I ) ∼ = Hom R ( I, R ) by (4). By applying the functorHom R ( I, − ) to 0 → R x −→ R → R → 0, we obtain Hom R ( I, R ) ∼ = Hom R ( I, R ) ∼ =Hom R ( I, R ). Here, we have I = I/xI = I/ (( x ) ∩ I ) ∼ = ( I + ( x )) / ( x ) = IR .Hence, it is enough to show that we have the natural isomorphism Hom R ( I, I ) ∼ =Hom R ( IR, IR ).By applying the functor Hom R ( I, − ) to 0 → I x −→ I → IR → 0, we get0 → Hom R ( I, I ) x −→ Hom R ( I, I ) → Hom R ( I, IR ) → Ext R ( I, I ) x −→ Ext R ( I, I ) . (5) n the other hand, by applying the functor Hom R ( I, − ) to (3), we have Hom R ( I, R/I ) ∼ =Ext R ( I, I ) by (4). It follows that x is a non-zerodivisor of Ext R ( I, I ) sinceAss R (Ext R ( I, I )) = Supp R I ∩ Ass R ( R/I ) ⊆ Ass R ( R/I ) . Hence (5) provides the isomorphism Hom R ( I, I ) ∼ = Hom R ( I, IR ) ∼ = Hom R ( IR, IR )as desired. (cid:3) Let us prove Theorem 4.4. Proof of Theorem 4.4. (a) ⇔ (b) follows from Theorem 2.13, and (c) ⇔ (d) followsfrom Proposition 4.6. Thus we have only to shows that (a) ⇔ (c).(a) ⇒ (c): By Corollary 2.15, we may assume that A is generically Gorenstein.For each p ∈ Min R , we have P = p × M ∈ Min A , whence A P ∼ = R p ⋉ M p isGorenstein. Hence, by Fact 4.3 (c), M p ∼ = R p since M is faithful. It follows that M is of rank one. Since M ∼ = M ∗∗ is torsionfree, M ∼ = I for some ideal I of R . We mayassume that M = I . Then A = R ⋉ I . Set K = ( R : I ) × R as an R -module. For ( a, x ) ∈ A and ( b, y ) ∈ K , let us define an A -action into K asfollows: ( a, x ) ◦ ( b, y ) = ( ab, ay + bx ) . With this action, K is an A -module. It is standard to show that K ∼ = Hom R ( A, R ) ∼ = ω A as A -modules. Furthermore we have A ⊆ K ⊆ Q( A ) = Q( R ) × Q( R ), thus K is afractional canonical ideal of A .On the one hand, we get ω ∗ A ∼ = A : K = ( R ⋉ I ) : (( R : I ) × R ) = I × ( I : ( R : I )) , where the last equality follows from the following argument. Let ( a, x ) ∈ Q( A ) =Q( R ) × Q( R ). Then( a, x ) ∈ ( R ⋉ I ) : (( R : I ) × R ) ⇔ ( a, x )( b, y ) = ( ab, ay + bx ) ∈ R ⋉ I for all b ∈ R : I and all y ∈ R ⇔ ( a ∈ R : ( R : I ) = I,x ∈ I : ( R : I ) . On the other hand, we gettr A ( ω A ) =( A : K ) K =( I × ( I : ( R : I )))(( R : I ) × R )=( R : I ) I × ( I + ( R : I )( I : ( R : I )))=( R : I ) I × I. Hence tr A ( ω A ) ∼ = ω ∗ A shows that I ∼ = ( R : I ) I . It follows that M = I ∼ = tr R ( I ).(c) ⇒ (a): Let M ∼ = I for some trace ideal I of R . Then M p ∼ = I p = R p for all p ∈ Min R since M is faithful. By noting that Min A = { p × M | p ∈ Min R } since × M ) = 0, A is generically Gorenstein. We may assume that M = I . Then, bythe same argument of the proof of (a) ⇒ (c), K = ( R : I ) × R is a fractional canonicalideal of A . Hence, to prove tr A ( ω A ) ∼ = ω ∗ A , it is enough to show I = ( R : I ) I and I : ( R : I ) = I by the proof of (a) ⇒ (c).We obtain that ( R : I ) I = I by Remark 2.12 (b). Furthermore, by Remark 2.12(b), we have I ⊆ I : ( I : I ) = I : ( R : I ) ⊆ R : ( R : I ) = I . Therefore, we havetr A ( ω A ) = ( R : I ) I × I = I × ( I : ( R : I )) ∼ = ω ∗ A as desired.When this is the case, assume that A is not Gorenstein. Let I be the trace idealisomorphic to M . In the computation of r( A ), we may assume that M = I . Choosea canonical ideal ω ( A and a ∈ A so that ω = aω and a ∈ ω . Then A/ω isGorenstein of dimension d − x , . . . , x d − ∈ m sothat x , . . . , x d − is an R , R/I , A , and A/ω -sequence. Then, by Proposition 4.1 andProposition 4.7, we can pass to R → R/ ( x , . . . , x d − ). Thus we may assume that d = 1. In the case of dimension one, our assertion follows from [10, Proposition6.5]. (cid:3) Corollary 4.8. Let ( R, m ) be a Gorenstein local domain of dimension d > and M a nonzero maximal Cohen-Macaulay R -module. Set A = R ⋉ M . Then the followingassertions are equivalent: (a) tr A ( ω A ) ∼ = ω ∗ A ; (b) can . red A ≤ ; (c) M is isomorphic to some trace ideal of R ; (d) M ∗ is isomorphic to some finite birational extension B of R such that B M is aCohen-Macaulay local ring of dimension d for all M ∈ Max B .When this is the case, if A is not Gorenstein, then r( A ) = r( R/I ) + 2 where I denotes the trace ideal isomorphic to M .Proof. Let M be a nonzero maximal Cohen-Macaulay R -module. Then, since M ∼ = M ∗∗ , M can be embedded into a finitely generated free R -module F . Hence, if aM = 0 for a ∈ R , then a = 0 since R is a domain. (cid:3) We close this paper with examples of Corollary 4.8 arising from semigroup rings. Definition 4.9. Let a , a , . . . , a ℓ ∈ Z n ( ℓ > 0) be lattice points. Set C = h a , a , . . . , a ℓ i = ( ℓ X i =1 c i a i | ≤ c i ∈ Z for all 1 ≤ i ≤ ℓ ) and call it the semigroup generated by a , a , . . . , a ℓ . Let S = k [[ X , . . . , X n ]] be theformal power series ring over a field k . We then set k [[ C ]] = k [[ X a , X a , . . . , X a ℓ ]]in S , where X a = X a X a · · · X a n n for a = ( a , a , . . . , a n ). The ring k [[ C ]] is calledthe semigroup ring of C over k . roposition 4.10. Let a , a , . . . , a ℓ ∈ Z n ( ℓ > be lattice points. For a positiveinteger m , set R m = k [[ X m a , X m a , . . . , X m a ℓ ]] . Suppose that R is a Gorenstein normal domain of dimension n . Then R m is aGorenstein domain and the integral closure of R m is R . Furthermore, if m = ab with some positive integers a and b , then R m ⊆ R a ⊆ R and R a is a finitely generated R m -module. Therefore, the canonical reduction numberof R m ⋉ Hom R m ( R m , R a ) is two.Proof. Note that the k -algebra homomorphism k [[ X , . . . , X n ]] → k [[ X , . . . , X n ]],where X i X mi for 1 ≤ i ≤ n , provides the isomorphism R ∼ = R m of rings. Thus R m is Gorenstein. Since ( X a j ) m ∈ R m for all 1 ≤ j ≤ ℓ , we get R m ⊆ R ⊆ R m ⊆ R , where R denotes the integral closure of R . Thus R m = R since R is normal.Suppose that m = ab . Then we have R m ⊆ R a ⊆ R and R a is finitely generatedas an R m -module by the following claim. (cid:3) Claim 1. R a = X c = c ( a a )+ ··· + c ℓ ( a a ℓ ) for ≤ c , c , . . . , c ℓ < b R m X c Proof of Claim 1. The inclusion ⊇ is clear, thus we have only to show the inclusion ⊆ . Let X c ∈ R a . Write c = c ( a a ) + · · · + c ℓ ( a a ℓ ), where 0 ≤ c , c , . . . , c ℓ . Thenwe can find integers r j and 0 ≤ q j < b such that c j = br j + q j for all 1 ≤ j ≤ ℓ .Hence we get c = ℓ X j =1 c j a a j = ℓ X j =1 ( br j + q j ) a a j = ℓ X j =1 ( r j m a j + q j a a j ) . It follows that X c = (cid:16) X P ℓj =1 r j m a j (cid:17) (cid:16) X P ℓj =1 q j a a j (cid:17) is in the right hand side of Claim1. (cid:3) Example 4.11. For m > 0, let R m = k [[ X m , X m Y m , X m Y m ]] be a semigroup ringover a field k . 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Department of Mathematics and Informatics, Graduate School of Science andEngineering, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba, 263-8522, Japan E-mail address : [email protected]@chiba-u.jp