/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 4, 2015, 458-461 ISSN 1307-5543 – www.ejpam.com A Note on Prüfer ⋆-multiplication Domains Olivier A. Heubo-Kwegna Department of Mathematical Sciences, Saginaw Valley State University, University Center MI 48710, USA Abstract. In this note, we prove that for an arbitrary star operation ⋆ on a domain R, the domain R is a Prüfer ⋆-multiplication domain if every 2-generated ideal of R is ⋆ f -invertible. Some characterizations of Prüfer-⋆ multiplication domains are therefore obtained. 2010 Mathematics Subject Classifications: 13A15, 13A18, 16W50 Key Words and Phrases: Star operation; ⋆-ideal; Prüfer ⋆-multiplication domain 1. Introduction Throughout this note R denotes an integral domain with quotient field K . LetF (R) be the set of all nonzero fractional ideals of R and f (R) be the set of all nonzero finitely generated fractional ideals of R. A star operation on R is a mapping A→ A⋆ ofF (R) intoF (R) such that for all A, B ∈ F (R) and for all a ∈ K \ {0}, (i) (a)⋆ = (a) and (aA)⋆ = aA⋆; (ii) A⊆ A⋆ and A⊆ B⇒ A⋆ ⊆ B⋆, and (iii) A⋆⋆ := (A⋆)⋆ = A⋆. For an overview of star operations, the reader may refer to [5, Sections 32 and 34]. Given a star operation ⋆ on R, one can construct a new star operation ⋆ f as follows: for each A∈ F(R), A⋆ f = ∪{B⋆|B ⊆ A and B ∈ f (R)}. A star operation is said to be of finite type if ⋆ f = ⋆. Since (⋆ f ) f = ⋆ f , ⋆ f is a finite type star operation for any given star operation ⋆ on R. Note that d f = d, where d is the identity star operation and if ⋆ is the v-operation we denote v f := t and call it the t-operation. A nonzero ideal A of R is a ⋆-ideal if A⋆ = A. Similarly, we call a ⋆-ideal of R a ⋆-prime ideal of R if it is also a prime ideal. We call a maximal element in the set of all proper ⋆-ideals of R a ⋆-maximal ideal of R. We denote Spec⋆(R) the set of all ⋆-prime ideals Email address: oheubokw@svsu.edu http://www.ejpam.com 458 c© 2015 EJPAM All rights reserved. O. Heubo-Kwegna / Eur. J. Pure Appl. Math, 8 (2015), 458-461 459 of R and Max⋆(R) the set of all ⋆-maximal ideals of R. An A ∈ F (R) is said to be ⋆-invertible if (AA−1)⋆ = R, whereas a domain R is a Prüfer ⋆-multiplication domain (in short, P⋆MD) if every finitely generated ideal A of R is ⋆ f -invertible, i.e., (AA−1)⋆ f = R for any A∈ f (R). Thus a Prüfer domain is a PdMD and PvMD is often called a Prüfer multiplication domain. Many authors have previously produced several characterizations of Prüfer-⋆ multiplica- tion domains (for instance see [1–3, 6]). The aim of this note is to provide some new charac- terizations of Prüfer-⋆multiplication domains. We precisely show that a domain R is a P⋆MD if and only if each 2-generated ideal of R is ⋆ f -invertible. Note that this result is a generalization of the fact that a domain is Prüfer if and only if each 2-generated ideal is invertible [9, page 7]. We also show that a domain R is a P⋆MD if and only if (a) ∩ (b) is ⋆ f -invertible for all a, b ∈ R \ {0}. The latest result has also been shown in the v-domain context [8] and in the PvMD context [7]. 2. Main Results We start this section with the recollection of some facts about star operations. Let ⋆ be a star operation on R. Recall that ⋆ is stable if (A∩ B)⋆ = A⋆∩ B⋆ for all A, B ∈ F (R). Now define e⋆ by Ae⋆ := ∩{ARM |M ∈ Max⋆ f (R)}, for all A ∈ F (R). Then it is well known that e⋆ is a stable star operation on R of finite type called the stable star operation of finite type associated to ⋆. It is not hard to see that Maxe⋆(R)=Max⋆ f (R)[4, Corollary 3.5(2)]. From the latest fact, it then follows that an ideal A is e⋆-invertible if and only if it is ⋆ f -invertible (in fact, if a star operation ⋆ is of finite type, then (AA−1)⋆ = R if and only if AA−1 6⊆ M for all M ∈ Max⋆(R)). From this observation it then follows that P⋆MD, P⋆ f MD, and Pe⋆MD coincide. Lemma 1. Let A be a finitely generated ideal of R and ⋆ a star operation on R. If A is ⋆ f -invertible, then ARM is principal for every M ∈ Max⋆ f (R). Proof. Suppose that A is ⋆ f -invertible. From the above observation, it follows that A is e⋆-invertible, i.e., (AA−1)e⋆ = R. We have, for each maximal ⋆ f -ideal M , RM = (AA−1)e⋆RM = ⋂ {(AA−1)RN |N ∈Max⋆ f (R)}RM = (AA−1)RM [4, Lemma 2.4.(1)]. Thus ARM is invertible and therefore principal. Theorem 1. Let R be an integral domain and let ⋆ be a star operation on R. Then the following statements are equivalent for an integral domain R. (i) RM is a valuation domain for all M ∈ Max⋆ f (R). (ii) R is a P⋆MD. (iii) Every nonzero fractional finitely generated ideal of R is ⋆ f -invertible. (iv) Every nonzero fractional 2-generated ideal is ⋆ f -invertible. REFERENCES 460 Proof. For (i)⇔ (ii) (see [1, Corollary 1.2]). (ii) ⇒ (iii) and (iii) ⇒ (iv) are clear. So it remains to prove that (iv)⇒ (i). Let x , y ∈ R, note that if P is a prime ideal of R, we have xRP + yRP = (a, b)RP for some a, b ∈ R. But if P is a ⋆ f -maximal ideal of R then, by Lemma 1, (a, b)RP is principal, that is, RP is a valuation domain. Corollary 1. A domain R is a P⋆MD if and only if (a)∩ (b) is ⋆ f -invertible for all a, b ∈ R \ {0}. Proof. Note that we have (ab)−1[(a)∩ (b)] = (a, b)−1. So (ab)−1[(a) ∩ (b)](a, b) = (a, b)−1(a, b) and � (ab)−1[(a)∩ (b)](a, b) � ⋆ f = � (a, b)−1(a, b) � ⋆ f . Thus if a, b ∈ R \ {0}, (a)∩ (b) is ⋆ f invertible if and only if (a, b) is ⋆ f -invertible. Hence R is a P⋆MD if and only if (a)∩ (b) is ⋆ f -invertible for all a, b ∈ R \ {0} by Theorem 1(iv). Recall that a ⋆-ideal A of R is of finite type if A= (a1, . . . , an) ⋆ for some (0) 6= (a1, . . . , an) ⊆ A. Note that if ⋆= ⋆ f , then A⋆ is of finite type if and only if A⋆ = (a1, . . . , an) ⋆ for some (0) 6= (a1, . . . , an) ⊆ A. If ⋆ is a star operation of finite type, then a ⋆-invertible ideal is of finite type. Also note that from [5, Proposition 32.2(b)] and the fact that (z)⋆ = (z) for any z ∈ K , we have ((a)∩ (b))⋆ = (a) ∩ (b) for any star operation ⋆ on R. Thus (a)∩ (b) is a ⋆-ideal of R for all a, b ∈ R \ {0}. Corollary 2. Let R be an integral domain such that � (ab)−1[(a)∩ (b)](a, b) � ⋆ = R. Then R is a P⋆MD if and only if (a)∩ (b) is of finite type. Proof. Suppose that R is a P⋆MD. Then (a)∩(b) is ⋆ f -invertible by Corollary 1. So (a)∩(b) is of finite type following the above discussion. Conversely if we assume that (a) ∩ (b) is of finite type, then from � (ab)−1[(a)∩ (b)](a, b) � ⋆ = R, it follows that (a) ∩ (b) is ⋆ f -invertible and hence R is a P⋆MD by Corollary 1. Remark 1. Note that the preceding theorem and corollaries give new characterizations of Prüfer ⋆-multiplication domains which generalize some of the classical characterizations of Prüfer v- multiplication domains (see [7, Lemma 1.7, Corollary 1.8, and Corollary 1.9]). ACKNOWLEDGEMENTS The author wishes to express his gratitude to Bruce Olberding for bringing up the problem treated in this paper following some discussions on Prüfer-⋆multipli- cation domains. References [1] D. D. Anderson, D. F. Anderson, M. Fontana, and M. Zafrullah. On v-Domains and Star Operations, Communications in Algebra, 2, 141-145. 2008. [2] D. F. Anderson, M. Fontana, and M. Zafrullah. Some remarks on Prüfer ⋆-multiplication domains and class groups, Journal of Algebra, 319, 272-295. 2008. [3] G. W. Chang. 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