Weakly Krull Inside Factorial Domains

Transcription

1 Weakly Krull Inide Factorial Domain D. D. ANDERSON, The Univerity of Iowa, Department of Mathematic, Iowa City, Iowa 52242, MUHAMMED ZAFRULLAH, Idaho State Univerity, Department of Mathematic, Pocatello, ID , GYU WHAN CHANG, The Univerity of Incheon, Department of Mathematic, Incheon, Korea, 1 Introduction Chapman, Halter-Koch, and Kraue [8] introduced the notion of an inide factorial monoid and integral domain. Throughout we will confine ourelve to the integral domain cae, but the intereted reader may eaily upply definition and proof for the monoid cae. An integral domain D i inide factorial if there exit a divior homomorphim ϕ: F D = D {0} where F i a factorial monoid and for each x D,thereexitann 1 with x n ϕ(f ). They howed [8, Propoition 4] that an inide factorial domain may be defined in term of a Cale bai and it will be thi characterization that we ue for the definition of an inide factorial domain. A ubet Q D i a Cale bai for D if hqi = {uq α1 q αn u U(D), q αi Q} i a factorial monoid with prime Q and for each d D there exit an n 1 with d n hqi. Here U(D) i the group of unit of D. A domain D i inide factorial if and only if D ha a Cale bai. They howed [8, Theorem 4] that D i inide factorial if and only if D, the integral cloure of D, i a generalized Krull domain with torion t-cla group Cl t (D), foreachp X (1) (D), the valuation domain D P ha value group order-iomorphic to a ubgroup of (Q, +) (we ay that D i rational), and D D i a root extenion (i.e., for each x D, thereexitann 1 with x n D). Recall that an integral domain D i weakly Krull [4] if D = T P X (1) (D) D P where the interection i locally finite. While an integrally cloed inide factorial domain i weakly Krull, an inide factorial domain need not be weakly Krull (ee Example 3.2 below). The purpoe of thi paper i threefold. Firt we how (Theorem 3.1) that an inide factorial domain ha a unique minimal root extenion which i weakly Krull (and necearily inide factorial). Second, we give (Theorem 3.3) a number of characterization of weakly Krull inide factorial domain. For example, we how that for an inide factorial domain the following condition are equivalent: (1) D i weakly Krull, (2) D i an almot GCD domain, (3) t-dim D =1, and (4) for each element q of a Cale bai for D, (q) i primary. Third, Theorem 3.1 and 3.3 are

2 then ued to give a new proof that if D i inide factorial, then D i a rational generalized Krull domain with torion t-cla group and D D i a root extenion. 2 Preliminarie In thi ection we review ome reult on Cale bae, AGCD domain, and root extenion. Let D be an inide factorial domain with Cale bai Q. Now by definition the element of Q are prime in the monoid hqi and the map Q X (1) (D) given by q p (q) i a bijection [8, Theorem 2]. (In fact, if Q 0 D with bijection a above, then Q 0 i a Cale bai.) However, the element of Q need not be irreducible nor primary a element of D. Alo, recall that an integral domain D i an almot GCD domain (AGCD domain) [10]ifforx, y D,thereexitann 1 with x n D y n D, or equivalently (x n,y n ) v, principal. An AGCD domain D ha torion t-cla group [5, Theorem 3.4], D D i a root extenion [10, Theorem 3.1], and D i a PVMD with torion t-cla group [10, Theorem 3.4 and Corollary 3.8]. Now a PVMD with torion t- cla group i an AGCD domain [10, Theorem 3.9], but if D i an integral domain with D D a root extenion and D apvmdwithtoriont-cla group, or equivalently, D i an AGCD domain, D need not be an AGCD domain. The following example i [3, Theorem 3.1] (alo ee Example 3.2 of thi paper). Let K ( L be a purely ineparable field extenion and let D = K +(X)L[X] where X i a et of indeterminate with X > 1. ThenD ( D = L [X] i a root extenion and L[X] i a UFD (and hence an AGCD domain), but D i not an AGCD domain. Note that if X =1, D i an AGCD domain. The following reult on root extenion will be frequently ued. (1) [5, Theorem 2.1], [8, Propoition 5] Let R S be a root extenion of commutative ring. The map Spec(S) Spec(R) given by Q Q R i an order iomorphim and homeomorphim with invere given by P P = { S n P for ome n 1}. Moreover, R P S Q i a root extenion. (2) [8, Propoition 5] Let D S be a root extenion of integral domain. Then D i inide factorial if and only if S i inide factorial. Of coure, in the cae where D i inide factorial and S K, thequotient field of D, D S i a root extenion if and only if D S i integral. We have remarked that an inide factorial domain need not be weakly Krull. The previouly mentioned example D = K +(X)L[X] where K ( L ipurelyineparable and X > 1 i uch an example, ee Example 3.2 below. In [6, Propoition 2.4] we howed that the following condition are equivalent for an inide factorial domain D with Cale bai Q: (1) for each q Q, p (q) i a maximal t-ideal, (2) D i weakly Krull, (3) D i an AGCD domain, (4) each q Q i a primary element, and (5)ditinct element of Q are v-coprime. See Theorem 3.3 below where thee and everal more equivalence are given. 3 Weakly Krull Inide Factorial Domain We begin thi ection by howing that an inide factorial domain ha a unique minimal root extenion which i weakly Krull. We then give an example of an inide

3 factorial domain that i not weakly Krull and give a number of characterization of weakly Krull inide factorial domain. Theorem 3.1. Let D be an inide factorial domain and let D # = T P X (1) (D) D P. (1) D D # i a root extenion and hence D # i inide factorial. (2) D # i weakly Krull. (3) Let S be an integral domain containing D uch that S i weakly Krull and D S i a root extenion. Then D # S. (4) Suppoe that S i an integral domain with D # S a root extenion. Then S i a weakly Krull inide factorial domain. Proof. Here D i an inide factorial domain with quotient field K. Let Q = {q α } be a Cale bai for D and let P α = p (q α ).SoQ X (1) (D) given by q α P α i a bijection. (1) Let 0 6= x D #. Write x = a/b where a, b D. Since uitable power of a and b lie in hqi, wecanchooen 1 with x n = qα n1 1 qα n where U(D) and each n i Z. Suppoe that ome n i < 0. With a change of notation, if neceary, we can aume that n 1,,n i < 0 and n i+1,,n > 0. Now x n D Pαi,o qα n1 1 qα n = r/t where r D and t D P αi. Then tqα ni+1 i+1 qα n = rqα n1 1 qα ni i P αi. But no factor on the left hand ide lie in P αi, a contradiction. Hence each n i > 0, ox n D. Since D D # i a root extenion, D # i inide factorial. (2) For N X (1) (D # ),lets = D # N and P = N D. SinceD D # i a root extenion, P X (1) (D). Now D # D P give D # N (D P ) S = D P where the equality follow ince ht P P =1and P P S =. But certainly D P D # N,o D P = D # N. Thu D# T N X (1) (D # ) D# N = T P X (1) (D) D P = D # and hence D # = T N X (1) (D # ) D# N. Let 06= x K. Then x i a unit in almot all the D Pα. Thi follow ince 0 6= a D i in P α = p (q α ) if and only if a n hqi ha q α a one of it factor and thu a i in only finitely many P α. Since each D # N i equal to a unique D P α, x i a unit in almot all the D # N. Hence D# i weakly Krull. (3) Suppoe that D S i a root extenion and that S = T N X (1) (S) S N.ForN X (1) (S), P = N D X (1) (D); od P S N. Then D # = T P X (1) (D) D P T N X (1) (S) S N = S. (4) Suppoe that D # S i a root extenion. Then D S i alo a root extenion. Hence Q i a Cale bai for D # and S. Since D # i weakly Krull, p q α D # X (1) (D # ) give that q α D # i p q α D # -primary. We claim that q α S i q α S- primary. Suppoe that xy q α S and x/ q α S. Chooe n 1 with x n,y n D # and x n y n q α D #. Then x n / p q α D #,oforomem 1, (y n ) m q α D # q α S. Thu S ha a Cale bai Q = {q α } with each q α S primary. By the remark preceding Theorem 3.1, S i a weakly Krull inide factorial domain. Of coure, for D inide factorial, D = D # if and only if D i weakly Krull. Note that by [8, Propoition 5(f)] D i tamely inide factorial if and only if D # i. We next give an example of an inide factorial domain that i not weakly Krull.

4 Example 3.2. Let D = K +(X)L[X] where K L i a purely ineparable field extenion and X i a nonempty et of indeterminate over L. ThenD D = L[X] i a root extenion ince K L i purely ineparable and D being factorial i inide factorial. Thu D i inide factorial. Alternatively, if we take Q to be a complete et of nonaociate prime element of K[X], thenq i a Cale bai for D and hence D i inide factorial. Suppoe that X =1.Thendim D =1,oD = D # and hence D i weakly factorial. Suppoe that X > 1. We claim that D # = L X. Let P be a height-one prime ideal of D; op = fl[x] D where f L[X] i irreducible. Chooe g D P with g(0)=0. ( If X =1and f = X, we can t chooe uch a g. But uppoe X > 1. If f X, takeanyg X {f} while if f / X; take g = X X.) Now for a L, a = ag/g D P.HenceL D #,od # = L[X]. We next give a number of condition equivalent to D being a weakly Krull inide factorial domain. But firt we need ome more definition. An integral domain D i almot weakly factorial [4] if ome power of each nonunit of D i a product of primary element. By [4, Theorem 3.4], D i almot weakly factorial if and only if D i weakly Krull and Cl t (D) i torion. An integral domain D i a generalized weakly factorial domain [7] if each nonzero prime ideal of D contain a nonzero primary element. Thu an almot weakly factorial domain i a generalized weakly factorial domain. Two element x and y of an integral domain D are power aociate if there exit n, m 1 and a unit U(D) with x n = y m. Given a nonzero prime ideal P of D, b i a bae for P if P = p (b). Thu if (b) i P-primary, b i a bae for P. Finally, t-dim D =1if every prime t-ideal of D i a maximal t-ideal. Theorem 3.3. For an integral domain D the following condition are equivalent. (1) D i inide factorial and weakly Krull. (2) D i inide factorial and AGCD. (3) D i inide factorial and t-dim D =1. (4) D i inide factorial with a Cale bai Q uch that qd i a maximal t-ideal for each q Q. (5) D i inide factorial and for each Cale bai Q for D, qd i a maximal t-ideal for each q Q. (6) D i inide factorial with a Cale bai Q uch that each q Q i primary. (7) D i inide factorial and for each Cale bai Q for D and each q Q, q i primary. (8) D i inide factorial with a Cale bai Q uch that ditinct element of Q are v-coprime. (9) D i inide factorial and for each Cale bai Q for D ditinct element of Q are v-coprime. (10 ) D i almot weakly factorial and two nonzero primary element of D are either v-coprime or power aociate. (11 ) D i a weakly Krull AGCD domain and two nonzero primary element of D are either v-coprime or power aociate. (12 ) D i a generalized weakly factorial domain uch that for each height-one prime ideal P of D, every pair of bae element of P are power aociate and for each nonzero x P, there i an n 1 with x n = db where b i a bae for P and d/ P. Proof. The equivalence of (1), (2), and (4) (9) i given by [6, Propoition 2.4]. (3) (4) For D inide factorial, X (1) (D) ={ qd q Q} for any Cale bai Q

5 for D. Since a height-one prime ideal i a t-ideal, the equivalence follow. (6)= (10) Let x be a nonzero nonunit of D. Thenomepowerx n hqi. Sox n i a product of primary element and hence D i almot weakly factorial. Suppoe that q 1 and q 2 are nonzero primary element of D. If p (q 1 ) 6= p (q 2 ),then(q 1,q 2 ) v = D (for ay by (6)= (3) p (q 1 ) and p (q 2 ) are maximal t-ideal). So uppoe p (q 1 )= p p (q 2 )= (q) where q Q. Thenq n 1 1 = 1 q m1 and q2 n2 = 2 q m2 for ome n 1,n 2,m 1,m 2 1 and unit 1, 2. But then q 1 and q 2 are power aociate. (10)= (6) Since D i almot weakly factorial, D i weakly Krull and each height-one prime P α contain a P α -primary element q α. Let Q = {q α }. We claim that Q i a Cale bai for D. If 1 q n 1 α 1 q n α = 2 q m 1 α 1 q m α where 1, 2 are unit and n i,m i 0, then qα ni i D Pαi = 1 qα n1 1 qα n D Pαi = 2 qα m1 1 qα m D Pαi = qα mi i D Pαi ;on i = m i. Hence hqi i a factorial monoid. Let x be a nonzero nonuit of D. Then ome power x n i a product of primary element. But each primary element in thi factorization i power aociate to ome q α.thuraiingx n to an appropriate power give that ome power of x i in hqi. (11)= (10) It uffice to how that an AGCD weakly Krull domain i almot weakly factorial. But we have already remarked that an AGCD domain ha torion t-cla group and that a domain i almot weakly factorial if and only if it i a weakly Krull domain with torion t-cla group. (6)= (11) Since (6)= (1), D i weakly Krull and ince (6)= (2), D i an AGCD domain. And by (6)= (10) two nonzero primary element are either v-coprime or power aociate. (10)= (12) Certainly an almot weakly factorial domain i a generalized weakly factorial domain. Now an almot weakly factorial domain i weakly Krull and in a weakly Krull domain an element x with p (x) =P 0, P 0 a height-one prime, i primary (for (x) =x T P X (1) (D) D P = T P X (1) (D) xd P = xd P 0 D). Thu two bae for a height-one prime ideal P are P-primary and hence by hypothei are power aociate. Finally, let 0 6= x P.SinceDi almot weakly factorial, ome x n i a product of primary element, ay x n = q 1 q i q i+1 q m where q 1,,q i P (with necearily i 1) andq i+1,,q m / P. Now (q 1 q i ) i till P-primary (for D i weakly Krull and p (q 1 q i )=P). Take b = q 1 q i and d = q i+1 q n ; o x n = db where d / P and b i a bae for P. (12)= (1) A generalized weakly factorial domain i weakly Krull [7, Corollary 2.3]. For each P X (1) (D) chooe a bae element x P for P. We claim that Q = {x P } i a Cale bai for D; and hence D i inide factorial. A in the proof of (10)= (6), hqi i a factorial monoid. Let x be a nonzero nonunit of D. LetP 1,,P m be the height-one prime ideal of D containing x. By hypothei, there i an n 1 o that x n = db where b i a bae for P 1 and d/ P 1.Nowband x P1 are power aociate, ay b r = ux P 1 where r, 1 and u i a unit. Then x nr = d r ux P 1 = x 1 x P 1 where x 1 / P 1.NowP 2,,P m are the height-one prime ideal containing x 1. By induction, we have x t 1 = βx2 P 2 x m P m where β i a unit and t, 2,, m 1. Hencex tnr hqi. 4 Micellaneou Reult We have already remarked that an integral domain D i inide factorial if and only if D i a rational generalized Krull domain with torion t-cla group and D D i a root extenion [8]. We begin thi ection by giving an alternative proof of the implication (= ) baed on Theorem 3.1 and 3.3. We then dicu when certain extenion of an inide factorial domain are again inide factorial.

6 Theorem 4.1. [8, Theorem 4] An integral domain D i inide factorial if and only if D i a rational generalized Krull domain with torion t-cla group and D D i a root extenion. Proof. (= ) Suppoe that D i inide factorial. By Theorem 3.1, D D # i a root extenion and D # i a weakly Krull inide factorial domain. By Theorem 3.3 D # i an AGCD domain. Thu we can apply reult from [10] concerning AGCD domain. By [10, Theorem 3.1], D # D # = D i a root extenion. Hence D D i a root extenion. By [10, Theorem 3.4 and Corollary 3.8] D i a PVMD with torion t-cla group. But by Theorem 3.1 again, D i weakly Krull. Hence D i a generalized Krull domain. It remain to how that for each P X (1) (D), the value group of D P i iomorphic to a ubgroup of (Q, +). Let Q be a Cale bai for D with P = p (p) where p Q. For 0 6= x K, there i a natural number n with x n = p n0 q n1 1 qn where n 0,,n Z, U(D) and q 1,,q Q. But q 1,,q are unit in D P,owecanwritex n = 0 p n 0 where 0 i a unit in D P.If v: K (R, +) i the valuation for D P,thennv(x) =v(x n )=v( 0 p n0 )=n 0 v(p). Hence v(x) = n0 n v(p). Thuim v i order-iomorphic to (Q, +). ( =) [8, Theorem 4] We end by conidering extenion of inide factorial domain. Let D be inide factorial. Then each overring E of D contained in D i again inide factorial (for D E i a root extenion). Moreover, by Theorem 3.1 if D i weakly Krull, o i E. Next uppoe that S i a multiplicatively cloed ubet of D. Then D D a root extenion give that D S D S = D S i a root extenion, D S i a rational generalized Krull domain, and D S ha torion t-cla group ince D doe [2, Theorem 4.4]. Thu D S i inide factorial. Alternatively, oberve that if Q i a Cale bai for D, then {q Q qd S 6= D S } i a Cale bai for D S. Moreover, if 6= Λ X (1) (D), then R = T P Λ D P i a weakly Krull inide factorial domain. For if we et S = ª ³ T q Q qd / Λ,then(D # ) S = P X (1) (D) D P = T P Λ D P = R. S We next how that D[X] i inide factorial if and only if D i inide factorial and D[X] D[X] i a root extenion. Alo ee [9, Theorem 3.2]. Now D[X] i inide factorial if and only if D[X] D[X] i a root extenion and D[X] i a rational generalized Krull domain with torion t-cla group. But D[X] i a rational generalized Krull domain with torion t-cla group if and only if D i (for D i a generalized Krull domain if and only if D[X] i, the natural map Cl t (D) Cl t (D[X]) i an iomorphim, and for N X (1) (D[X]), D[X] N i a DVR if N D = 0 and if N D 6= 0,thenD[X] N = D N D (X) ha the ame value group a D N D ) and D[X] D[X] a root extenion force D D to be a root extenion. However, D D a root extenion need not imply the D[X] D[X] i a root extenion. Let D = Z[ 5]. ThenD ( D i a root extenion and D i an AGCD (= weakly Krull) inide factorial domain, but D[X] ( D[X] i not a root extenion [1, Example 3.6]. Hence D[X] i not inide factorial. According to [1, Theorem 3.4], for an integral domain D, D[X] i an AGCD domain if and only if D i an AGCD domain and D[X] D[X] i a root extenion. Hence if D[X] i inide factorial, D[X] i AGCD (= weakly Krull) if and only if D i.

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