ROSE リポジトリいばらき ( 茨城大学学術情報リポジトリ )

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1 ROSE リポジトリいばらき ( 茨城大学学術情報リポジトリ ) TitleOn Characterizations of Divided Mon Author(s) OKABE, Akira Citation Mathematical Journal of Ibaraki Uni Issue Date 2002 URL Rights このリポジトリに収録されているコンテンツの著作権は それぞれの著作権者に帰属します 引用 転載 複製等される場合は 著作権法を遵守してください お問合せ先 茨城大学学術企画部学術情報課 ( 図書館 ) 情報支援係

2 Math. J., Ibaraki Univ., Vol. 34, 2002 On Characterizations of Divided Monoids AKIRA OKABE* 1. Introduction. Let G be a torsion-free abelian additive group, and let S be a subsemigroup of G which contains 0. Then S is called a grading monoid (cf. [8]). In this paper,unless otherwise indicated, we will call a grading monoid as a monoid. A of S and is denoted by q(s). Note that q(s) is evidently an abelian additive group. Each monoid which lies between S and q(s) is called an overmonoid of S. Before discussing divided monoids, let us recall some fundamental results and notions in monoid theory for the convenience of the reader. Remark 1. Let P be an ideal of S. Then P is a prime ideal of S if and only if T=S/P is an additive system of S. In particular, if P is a prime ideal of S and T=S/P, then the quotient monoid ST is usualy denoted by SP. the set of units of S, then the set M=S/U is an ideal of S that contains all the proper ideals of S and is called the maximal ideal of S. We shall often denote the maximal ideal of S by M(S). Remark 2. If P is a prime ideal of S, then PSP is the maximal ideal of SP. In Received December 25, 2001; revised May 7, Mathematics Subject Classification. Primary 20M12, Secondary 20M14. *Department of Mathematics, Oyama National College of Technology, 771, Nakakuki, Oyama, Tochigi, , Japan.

3 Akira OKABE 2. Divided Monoids. shall introduce the notion of a divided monoid as follows: A monoid S is called a divided monoid if every prime ideal P of S is a comparable ideal. Lemma 1. Let P be a prime ideal of a monoid S. Then the following conditions are equivalent. (1) P is a comparable ideal of S. (2) PSP=P. We shall denote the set of prime ideals of S by Spec(S). Lemma 2 ([6,Proposition 1(1)]). If S is a divided monoid, then Spec(S) is a linearly ordered set. Theorem 3. Let S be a monoid with quotient group G. Then the following conditions are equivalent: (1) S is a divided monoid. (2) For each prime ideal P of S, P:P=PSP:PSP.

4 On Characterizations of Divided Monoids

5 Akira OKABE Definition 5. A monoid S is called a Noetherian monoid if each ideal of S is finitely generated. Definition 6. Let T be a overmonoid of a monoid S. An element t of T is said over S if each element of T is integral over S. Let S be the set of elements of q(s) which are integral over S. Then S is an overmonoid of S. In fact, if We give a generalization theorem of Lemma 14 of [6]. Proposition 4. If I is a finitely generated fractional ideal of a monoid S, then I:I is integral over S. Proposition 6. Let T be an overmonoid of a monoid S. Assume that T is integral over S. Then (1) If P is a prime ideal of S, then there is a prime ideal Q of T such that

6 On Characterizations of Divided Monoids Theorem 7. If S is a Noetherian divided monoid, then dim(s)=1. This Theorem 7 is a Corollary of Theorem 15 in [6]. 3. Pseudo valuation monoids. In [8], we introduced the notion of a pseudo valuation monoid (for short, a PVM ) which is a semigroup version of a pseudo-valuation domain in [4] and studied some fundamental properties of a PVM. Definition 8. A monoid S with quotient group G is called a pseudo valuation monoid (for short, a PVM ), if each prime ideal P of S is G-strong. If a prime ideal P of S is G-strong, then P is also called a strongly prime ideal of S. Proposition 8 (cf. [10, Theorem 4 and Theorem 10] ). Let S be a monoid with maximal ideal M and quotient group G. Then the following statements are equivalent: (1) S is a PVM. (4) M is a strongly prime ideal of S. (5) S has the unique valuation overmonoid V with maximal ideal M. (6) There exists a valuation overmonoid V such that every prime ideal of S is also a prime ideal of V. In this Section we will give some new characterizations of a PVM and we will also show that each PVM is a divided monoid.

7 Akira OKABE We now give some new characterizations of a PVM. Theorem 10. Let S be a monoid with maximal ideal M and quotient group G. Then the following statements are equivalent. (1) S is a PVM. Theorem 11 (cf. [6, Proposition 1(2)]). Every PVM is a divided monoid. REFERENCES [1] D. E. Dobbs, Divided rings and going down, Pacific J. Math. 67 (1976), [2] D. E. Dobbs, Coherence, ascent of going down, and pseudo-valuation domains, Houston J. Math. 4 (1978), [3] R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, [4] J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains, Pacific J. Math. 75 (1978), [5] J. R. Hedstrom and E. G. Houston, Pseudo-valuation domains (II), Houston J. Math. 4 (1978), [6] M. Kanemitsu, On divided commutative semigroups, Far East J. Math. sci. (FJMS) 3 (2001), [7] R. Matsuda and M. Kanemirsu, On seminormal semigroups, Arch. Math. 69 (1997), [8] D. G. Northcott, Lessons on rings, modules, and multiplicities, Cambridge University Press, [9] A. Okabe, Some ideal-theoretical characterizations of divided domains, Houston J. Math. 12 (1986), [10] A. Okabe, On pseudo-valuation monoids, Research Reports of Oyama National College of Technology, No. 32 (2000),