On an Extension of Half Condensed Domains

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 12, Number 6 (2016), pp Research India Publications On an Extension of Half Condensed Domains Chahn Yong Jung Department of Business Administration, Gyeongsang National University, Jinju 52828, Korea. Waseem Khalid Department of Mathematics, The University of Lahore, 9-KM Sahiwal Road, Pakpattan, Pakistan. Waqas Ahmad Department of Mathematics, The University of Lahore, 9-KM Sahiwal Road, Pakpattan, Pakistan. Waqas Nazeer Division of Science and Technology, University of Education, Lahore 54000, Pakistan. Shin Min Kang 1 Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea. 1 Corresponding author.

2 Abstract In this paper, we introduce and study a class of integral domains D characterized by the property that whenever 0 =p XY with X, Y ideals of D, there exist X,Y (tinvertible) ideals of D such that X X, Y Y and pd = (X Y ) t. We call them t-condensed domains. We show that every fraction ring of a t-condensed domain is t-condensed, every PVMD is a t-condensed domain, and an integrally closed conditionally well behaved t-condensed domain is a PVMD, that a t-condensed domain with every maximal ideal a t-ideal is a HC domain. We also show that a conditionally well behaved t-condensed domain D is locally condensed for each maximal t-ideal Q of D, that an integrally closed t-condensed domain is a P -domain, and that a t-condensed domain with zero t-class group is a condensed domain. AMS subject classification: Primary 13A15, Secondary 13F05. Keywords: Condensed domain, HC domain, Prüfer v-multiplication domain. 1. Introduction and preliminaries In [6], Anderson and Dobbs introduced the concept of a condensed domain, that is, a domain for which XY ={rs r X, s Y } for all ideals X, Y of D. This concept is used to characterize Bézout domains (resp., PID s; resp., valuation domains) in suitably larger classes of domains. They proved that if D is a Bézout domain then D is condensed. In particular all principal ideal domains and all valuation domains are condensed. In [5], Anderson et al. proved that a domain is integrally closed and condensed if and only if it is Bézout. They also showed that if D is a Noetherian quasi-condensed domain then depth(d) 1. In [12], Gottlieb proved that under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. He also introduced the concept of strongly condensed domain, that is, a domain D is called strongly condensed if for each pair X, Y of ideals of D, wehavexy = ry for some r X or XY = Xs for some s Y. He showed that if a domain D contains an element of value 2 then D is strongly condensed. In [2], Anderson and Dumitrescu showed that a Noetherian domain D is condensed if and only if Pic(D) = 0 and D is locally condensed. A local domain is strongly condensed if and only if it has two-generated property. An integrally closed domain D is strongly condensed if and only if D is Bézout and generalized Dedekind domain with at most one maximal ideal of height greater than one. In [3], Anderson and Dumitrescu introduced the concept of half condensed domain (HC domain) and a strongly half condensed domain (SHC domain). A domain D is HC if whenever 0 =p XY with X, Y ideals of D, there exist X,Y (t-invertible) ideals of D such that X X, Y Y and pd = (X Y ) t. And D is SHC if whenever X, Y are nonzero ideals of D, XY = X 1 Y for some invertible ideal X 1 X or XY = XY 1 for some invertible ideal Y 1 Y. They proved that each overring of a HC domain is also

3 On an Extension of Half Condensed Domains 5311 HC and an integrally closed domain is SHC if and only if it is a generalized Dedekind domain with at most one maximal ideal of height greater than one. In this paper, we study the following extension of the concept of HC domain. We call a domain Dt-condensed if whenever 0 = p XY with X, Y ideals of D, there exist X,Y (t-invertible) ideals of D such that X X, Y Y and pd = (X Y ) t.we show that if D is t-condensed domain and L is a t-linked overring of D, then L is also a t-condensed domain. In particular every fraction ring of a t-condensed domain is t- condensed (Proposition 2.2). A domain D in which each pair of finitely generated ideals of D is a t-condensed pair is t-condensed. In particular every PVMD is a t-condensed domain (Proposition 2.4). If D is a t-condensed domain in which every maximal ideal is a t-ideal, then D is HC domain (Proposition 2.6). If D is a conditionally well behaved t- condensed domain, then D P is condensed for each maximal t-ideal P of D (Proposition 2.8). An integrally closed t-condensed domain is a P -domain (Proposition 2.10). If D is a t-condensed domain with Cl t (D) = 0, then D is condensed (Proposition 2.9). A domain D is a PVMD if and only if it is integrally closed conditionally well behaved and t-condensed domain (Proposition 2.11). This new concept of t-condensed domain depends upon the notion of star operation, v-operation, t-operation. A reader in need of a quick review on this topic may consult Sections 32 and 34 of Gilmer s book [11]. For the reader s convenience, we give a working introduction here for the notions involved. Let D be an integral domain with quotient field K and let F(D)denote the set of nonzero fractional ideals of D. A function A A : F(D) F(D)is called a star operation on D if satisfies the following three conditions for all 0 =a K and for all I,J F(D): (1) D = D and (ai) = ai, (2) I I and if I J then I J, (3) (I ) = I. An ideal I F(D) is called a -ideal if I = I. For all I,J F(D),wehave (IJ ) = (I J) = (I J ). These equations define the so-called -multiplication. If {I α } is a subset of F(D)such that I α =0, then I α is a -ideal. Also, if {I α} is a subset of F(D)such that ( ) ( ) I α is a fractional ideal, then Iα = I. α A star operation is said to be a stable star operation if (I J) = I J for all I,J F(D). The function f : F(D) F(D) given by I f = J, where J ranges over all nonzero finitely generated sub-ideals of I, is also a star operation; is said to be a star operation of finite character if = f. Clearly ( f ) f = f. Let max(d) denote the set of maximal -ideals, that is, ideals maximal among proper integral -ideals of D. Every maximal -ideal is a prime ideal. If is of finite character, then every proper -ideal is contained in some maximal -ideal, and is stable if and only if I = ID P for all I F(D), cf. [1, Corollary 4.2]. P max (D)

4 5312 C. Y. Jung, W. Khalid, W. Ahmad, W. Nazeer and S. M. Kang A -ideal I is of finite type if I = (a 1,...,a n ) for some a 1,...,a n I. An ideal I F(D) is said to be -invertible if (II 1 ) = D, where I 1 = (D : I) ={x K xi D}. If is of finite character, then I is -invertible if and only if II 1 is not contained in any maximal -ideal of D; in this case I = (a 1,...,a n ) for some a 1,...,a n I. Let 1, 2 be star operations on D. We write 1 2 if I 1 I 2 for all I F(D). In this case we get (I 1 ) 2 = I 2 = (I 2 ) 1 and every 1 -invertible ideal is 2 -invertible. Some well-known star operations are: the d-operation (given by I I), the v-operation (given by I I v = (I 1 ) 1 ) and the t-operation (defined by t = v f ). For every I F(D),wehaveI I t I v ;soav-ideal is a t-ideal. Recall from [9] that the quotient group of t-invertible fractional t-ideals modulo the subgroup of principal fractional ideals is called the t-class group denoted by Cl t (D). Let S be a subring of an integral domain D. Then D is t-linked over S if for each finitely generated fractional ideal A of S with A 1 = S,wehaveAD 1 = D [4]. A prime t-ideal P of a domain D is called well behaved if PD P is also a t-ideal [16]. A domain D is called well behaved domain if every prime t-ideal of D is well behaved [16]. A domain D is called a Krull domain if every nonzero ideal of D is t-invertible [15]. A domain D is called a Prüfer v-multiplication domain (PVMD) if every nonzero finitely generated ideal is t-invertible [14]. A domain D is called a P -domain if for each associated prime P of a principal ideal, D P is a valuation domain [14]. A Mori domain is an integral domain satisfying the ascending chain condition on integral divisorial ideals [8]. Throughout this paper, all rings are (commutative unitary) integral domains. For a domain D, denote the quotient field of D by K. Our standard reference for any undefined notation or terminology is [11]. 2. Main results Definition 2.1. A domain D is said to be a t-condensed domain if whenever 0 =p XY with X, Y ideals of D, there exist X,Y (t-invertible) ideals of D such that X X, Y Y and pd = (X Y ) t. Proposition 2.2. If D is a t-condensed domain and L is a t-linked overring of D. Then L is also a t-condensed domain. In particular every fraction ring of a t-condensed domain is t-condensed. Proof. Let X, Y be two non zero ideals of L and 0 =p XY. Then p = x 1 y x n y n for some x i X and y i Y.Take0 =d D with dx i,dy i D for all i. Then d 2 p (dx 1 D + +dx n D)(dy 1 D + +dy n D). Since D is t-condensed, d 2 p = (X Y ) t for some ideals X dx 1 D + +dx n D and Y dy 1 D + +dy n D of D. Now X = X /d x 1 D + +x n D, Y = Y /d y 1 D + +y n D

5 On an Extension of Half Condensed Domains 5313 are fractional ideals of D and pd = (X Y ) t. Since D = ( 1 p X Y ) t and L is t-linked, L = ( 1 p X Y L) t. Hence pl = ((X L)(Y L)) t with X L X and Y L Y. The in particular statement follows from [4, Proposition 2.3(3)]. Remark 2.3. Let D be a domain and X, Y two nonzero ideals of D. If X or Y is t- invertible, then X,Y is a t-condensed pair. Indeed, if Y is t-invertible and 0 =p XY, then pd = ((py 1 )Y ) t with py 1 X. Proposition 2.4. Every pair of finitely generated ideals of a domain D is a t-condensed pair if and only if D is t-condensed. In particular, every PVMD is t-condensed. Proof. For the first implication, let X, Y be any pair of ideals of D with 0 =p XY. Then p = x 1 y x n y n, where x k X and y k Y for k = 1,...,n. Now p X Y, where the ideals X and Y are given by X = (x 1,...,x n ) X and Y = (y 1,...,y n ) Y. By hypothesis, X,Y is a t-condensed pair, so there exist t-invertible ideals X X and Y Y such that pd = (X Y ) t. The converse implication is clear and the in particular assertion follows from Remark 2.3. Example 2.5. Consider the domain D = W[[Z 2,Z 3 ]], where W is a field. Then by [6, Example 2.3], D is condensed and hence t-condensed. It is clear that D is not PVMD because it is not integrally closed. Proposition 2.6. If D is a t-condensed domain having every maximal ideal a t-ideal of D, then D is HC. Proof. Let X, Y be two ideals of D and 0 =p XY. Since D is t-condensed, there exist ideals X X and Y Y such that pd = (X Y ) t, where X,Y are invertible ideals by given hypothesis. Hence pd = X Y with X X and Y Y. Example 2.7. By [3], a quasi-local HC domain is condensed. Proposition 2.6 shows that a quasi-local one-dimensional domain is t-condensed if and only if it is condensed. Therefore, the rings K + LK(U, V )[[L]] and K[[L 3,L 5,L 6,L 8,...]] are not t-condensed, cf. [6, Example 2.11]. Corollary 2.8. Let D be a t-condensed domain with Q a well behaved prime t-ideal of D then the localization D Q is condensed. In particular, if D is a conditionally well behaved t-condensed domain, then D M is condensed for each maximal t-ideal M of D. Proof. Since Q is well behaved, the unique maximal ideal QD Q is a t-ideal. By Propositions 2.2 and 2.6, D P is HC and hence condensed, cf. [3]. Proposition 2.9. Let D be a t-condensed domain with Cl t (D) = 0. Then D is condensed. Proof. Let X, Y be two ideals of D with 0 =p XY. Since D is t-condensed, there

6 5314 C. Y. Jung, W. Khalid, W. Ahmad, W. Nazeer and S. M. Kang exist X,Y (t-invertible) ideals such that X X, Y Y and pd = (X Y ) t. As X,Y are t-invertible ideals and Cl t (D) = 0, we get that X = xd and Y = yd for some x X X and y Y Y. Then pd = ((xd)(yd)) t = ((xy)d) t = (xy)d implies p = xyu for some unit element u D. Hence D is condensed. Proposition An integrally closed t-condensed domain is a P -domain. Proof. Let D be an integrally closed t-condensed domain and P an associated prime of a principal ideal. Then by Corollary 2.8, D P is condensed and by [5, Main Theorem] and [13, Theorem 63], D P is a valuation domain. In [3, Proposition 1.2], it is shown that an integral domain is Prüfer if and only if it is integrally closed and HC. We extend this result to t-condensed domains. Proposition A domain D is a PVMD if and only if it is an integrally closed conditionally well behaved and t-condensed. Proof. Clearly a PVMD is integrally closed and by Proposition 2.4 it is a t-condensed domain. By [9, Lemma 2.5], a PVMD is well behaved because in a PVMD the inverse of a finitely generated ideal is a v-ideal of finite type. Conversely, assume that D is integrally closed conditionally well behaved and t-condensed domain. Then by Corollary 2.8, [5, Main Theorem] and [13, Theorem 63], D M is valuation domain for each maximal t-ideal M of D. Hence D is PVMD. Corollary A domain D is an integrally closed t-condensed and a Mori domain if and only if D is Krull. Proof. For the first implication, because a Mori domain is well behaved [9, Lemma 2.5], Proposition 2.11 shows that D is a PVMD. Then by [7, Lemma 1.8] and [10, Corollary 6(b)], we get that D is a Krull domain. For the converse implication, it is clear that a Krull domain is integrally closed t-condensed and a Mori domain. In [3], the authors mentioned an alternative way to define HC pair. Let X 1,X 2 be a pair of nonzero ideals in a domain D. Then X 1,X 2 is a HC pair if and only if for each nonzero principal ideal X X 1 X 2, there exist invertible ideals X i X i for i = 1, 2 with X = X 1 X 2. [3, Proposition 1.6] shows that one could define an HC domain by replacing the word principal by the word invertible throughout the definition. Similarly, we mention an alternative way to define t-condensed pair. Let X 1,X 2 be a pair of nonzero ideals in a domain D. Then X 1,X 2 is a t-condensed pair if and only if for each nonzero principal ideal X X 1 X 2, there exist t-invertible ideals X i X i with X = (X 1 X 2 ) t. We note in the next proposition that we could have define a t-condensed domain by replacing principal ideal by invertible ideal.

7 On an Extension of Half Condensed Domains 5315 Proposition Let D be a t-condensed domain and X, Y 1,Y 2 are ideals of D with X invertible ideal and X Y 1 Y 2. Then X = (X 1 X 2 ) t for some t-invertible ideals X i with X i Y i for i = 1, 2. Proof. As X is invertible, XV = pd for some invertible ideal V and some p D. Then p AV (V Y 1 )Y 2. Since D is t-condensed, pd = (AB) t for some t-invertible ideals A, B with A VY 1 and B Y 2. Then X = V 1 (pd) = V 1 (AB) t. This implies X = (V 1 AB) t and AV 1 Y 1. So the desired property holds with X 1 = AV 1 and X 2 = B. Acknowledgement We would like to express our sincere thanks to The University of Lahore, Pakpattan Campus, Pakpattan for their support in this research work. References [1] D. D. Anderson, Star operations induced by overrings, Comm. Algebra, 16 (1988), pp [2] D. D. Anderson and T. Dumitrescu, Condensed domains, Canad. Math. Bull., 46 (2003), pp [3] D. D. Anderson and T. Dumitrescu, Half condensed domains, Houston J. Math., 30 (2004), pp [4] D. D. Anderson, E. G. Houston and M. Zafrullah, t-linked extensions, the t-class group and Nagata s theorem, J. Pure Appl. Algebra, 86 (1993), pp [5] D. F. Anderson, J. T. Arnold and D. E. Dobbs, Integrally closed condensed domains are Bézout, Canad. Math. Bull., 28 (1985), pp [6] D. F. Anderson and D. E. Dobbs, On the product of ideals, Canad. Math. Bull., 26 (1983), pp [7] S. El. Baghdadi, S. Gabelli and M. Zafrullah, Unique representation domains II, J. Pure Appl. Algebra, 212 (2008), pp [8] V. Barruci, On a class of Mori domains, Comm. Algebra 11 (1983), pp [9] A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Greek Math. Soc., 29 (1988), pp [10] T. Dumitrescu and M. Zafrullah, t-schreier domains, Comm. Algebra, 39 (2011), pp [11] R. Gilmer, Multiplicative Ideal Theory, Pure Appl. Math., vol. 12, Marcel Dekker Inc, New York, 1972.

8 5316 C. Y. Jung, W. Khalid, W. Ahmad, W. Nazeer and S. M. Kang [12] C. Gottlieb, On condensed Noetherian domains whose integral closures are discrete valuation rings, Canad. Math. Bull., 32(2) (1989), pp [13] I. Kaplansky, Commutative Rings, Chicago University, Chicago Press, [14] J. L. Mott and M. Zafrullah, On Prüfer v-multiplication domains, Manuscripta Math., 35 (1981), pp [15] J. L. Mott and M. Zafrullah, On Krull domains, Arch. Math., 56 (1991), pp [16] M. Zafrullah, Well behaved prime t-ideals, J. Pure Appl. Algebra, 65 (1990), pp