GCD AND LCM-LIKE IDENTITIES FOR IDEALS IN COMMUTATIVE RINGS

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1 GCD AND LCM-LIKE IDENTITIES FOR IDEALS IN COMMUTATIVE RINGS D. D. ANDERSON, SHUZO IZUMI, YASUO OHNO, AND MANABU OZAKI Absrac. Le A 1,..., A n n 2 be ideals of a commuaive ring R. Le Gk resp., Lk denoe he produc of all he sums resp., inersecions of k of he ideals. Then we have LnG2G4 G2 n/2 G1G3 G2 n/2 1. In he case R is an arihmeical ring we have equaliy. In he case R is a Prüfer ring, he equaliy holds if a leas n 1 of he ideals A 1,..., A n are regular. In hese wo cases we also have GnL2L4 L2 n/2 L1L3 L2 n/2 1. Relaed equaliies are given for Prüfer v-muliplicaion domains and formulas relaing GCD s and LCM s in a GCD domain generalizing gcda 1, a 2 lcma 1, a 2 a 1 a 2 are given. 1. Inroducion I is well known ha for naural numbers a 1 and a 2 or more generally for nonzero elemens of a PID In erms of ideals his says ha gcda 1, a 2 lcma 1, a 2 a 1 a 2. a 1 + a 2 a 1 a 2 a 1 a 2 or, since all ideals are principal, ha A 1 + A 2 A 1 A 2 A 1 A 2 for all ideals A 1, A 2 of a PID. In fac, his ideal equaliy nearly characerizes PIDs among inegral domains. Indeed, an inegral domain R is a Prüfer domain i.e., every nonzero finiely generaed ideal is inverible if and only if A 1 + A 2 A 1 A 2 A 1 A 2 for all ideals A 1, A 2 of R see Theorem 2.6. Dae: June 27, Mahemaics Subjec Classificaion. Primary: 13A15; Secondary: 13A05, 13F05. Key words and phrases. GCD, LCM, arihmeical ring, Prüfer ring, PVMD. 1

2 2 D. D. ANDERSON, S. IZUMI, Y. OHNO, AND M. OZAKI Less well known is he following formula for he gcd of n n 2 naural numbers a 1,..., a n in erms of heir lcms: GCD n gcda 1,..., a n lcma i1,..., a i2k a 1 a n 1 i 1 < <i 2k n 2 2k+1 n 1 i 1 < <i 2k+1 n lcma i1,..., a i2k+1 and he formula obained by inerchanging gcd and lcm: LCM n lcma 1,..., a n gcda i1,..., a i2k a 1 a n 1 i 1 < <i 2k n 2 2k+1 n 1 i 1 < <i 2k+1 n gcda i1,..., a i2k+1 see for example [9]. I is easy o prove ha he simple formula gcda 1,..., a n lcma 1,..., a n a 1 a n holds precisely when each prime divides a mos wo of he a i s. The corresponding ideal formulaion is as follows. Le R be a commuaive ring and le A 1,..., A n n 2 be ideals of R. For 1 k n pu Gk : Gk; A 1,..., A n A i1 + + A ik, Lk : Lk; A 1,..., A n A i1 A ik so G1 L1 A 1 A n, Gn A A n, Ln A 1 A n. Definiion 1.1. The ring R saisfies n for ideals A 1,..., A n of R n 2 if Gn L2k n L2k + 1 and saisfies n for ideals A 1,..., A n of R n 2 if Ln G2k n G2k + 1. Using he ceiling funcion and he floor funcion, we may express hese as follows: n GnL2L4 L2 n/2 L1L3 L2 n/2 1, n LnG2G4 G2 n/2 G1G3 G2 n/2 1.

3 GCD AND LCM-LIKE IDENTITIES FOR IDEALS 3 Noe ha 2 reduces o A 1 +A 2 A 1 A 2 A 1 A 2 and 2 reduces o A 1 A 2 A 1 + A 2 A 1 A 2. We are aking n 2 as he properies 1 and 1 are simply A 1 A 1 which is always rue. A commuaive ring R is called a chained ring resp., arihmeical ring if he laice of ideals of R is a chain resp. disribuive. Here a chain means a oally ordered se. So an inegral domain is a chained ring if and only if i is a valuaion domain. I is well known ha R is an arihmeical ring if and only if R M is a chained ring for each maximal ideal M of R. An inegral domain is a Prüfer domain if every nonzero finiely generaed ideal is inverible. R is a Prüfer domain if and only if R M is a valuaion domain for each maximal ideal M of R if and only if R M is a chained ring for each maximal ideal M of R. Thus a Prüfer domain is an arihmeical ring ha is an inegral domain. Finally, R is a Prüfer ring if every finiely generaed regular ideal is inverible. Here, an elemen is regular if i is no a zero-divisor and an ideal is regular if i conains a regular elemen. We show Theorem 2.4 if R is an arihmeical ring, hen n and n hold for all ideals A 1,..., A n of R and ha, if R is a Prüfer ring, hen n and n hold for all ideals A 1,..., A n of R when a leas n 1 of hem are regular. We also prove ha GCD n and LCM n hold for any GCD domain. Recall ha an inegral domain is a GCD domain if any wo elemens have a GCD, or equivalenly, any wo elemens have an LCM. We prove his in he more general seing of a Prüfer v-muliplicaion domain PVMD. To define a PVMD we need a lile back ground on he v- operaion and -operaion. Le R be an inegral domain wih quoien field K. For a nonzero fracional ideal I of R, I v : I 1 1 where I 1 {x K : xi R}. I is well known and easy o show ha I v {Rx : Rx I, x K}. Then I : {a 1,..., a n v : a 1,..., a n I \ {0}}. So, for I finiely generaed, we have I I v. A nonzero fracional ideal I is -inverible if II 1 R. Finally, R is PVMD if every nonzero finiely generaed ideal of R is -inverible. I is well known ha an inegral domain R is a GCD domain if and only if R is a PVMD in which every -inverible ideal is principal. Thus, for nonzero a 1,..., a n in a GCD domain, a 1,..., a n is a principal ideal. Suppose ha R is an inegral domain. Then nonzero a 1,..., a n R have an LCM if and only if a 1 a n is principal and in his case a 1 a n d where d lcma 1,..., a n. Noe ha LCM s and GCD s are only defined up o uni muliple. If a 1,..., a n is he

4 4 D. D. ANDERSON, S. IZUMI, Y. OHNO, AND M. OZAKI principal ideal d, hen he GCD exiss and gcda 1,..., a n d. If R is a GCD domain, hen gcda 1,..., a n d exiss and, since a 1,..., a n is principal, we have d a 1,..., a n. We show Theorem 2.4 ha if R is PVMD, Gn L2k L2k + 1, Ln G2k G2k + 1. Then aking R o be GCD domain and A 1 a 1,..., A n a n, we obain GCD n and LCM n for R Theorem 2.8. A good reference for muliplicaive ideal heory including Prüfer domains, GCD domains, and v-operaion is [5]. For resuls on Prüfer domains and Prüfer rings, see [8]. For a good survey of he v-operaion and -operaion, -inveribiliy, and PVMDs, see [10]. Finally, see [1] for a survey of GCD domains and relaed opics including PVMDs. Thus neiher n nor n always holds. In [2], however, we show ha he one-sided inclusion Ln G2k n G2k + 1 holds for general commuaive rings which may no have an ideniy. Indeed, his holds no only for ideal laices of rings bu in he quie general seing of a commuaive muliplicaion laice. Finally, in Secion 3 we give some examples o illusrae resuls from Secion Formulas n and n for ideals In his secion we prove various resuls concerning n and n for ideals of commuaive rings ha are menioned in he Inroducion. We begin wih he following fundamenal lemma. Lemma 2.1. Le R be a commuaive ring no necessarily wih ideniy and le A 1,..., A n n 2 be ideals of R. Suppose ha {A 1,..., A n } has a maximum resp., minimum elemen. Then n resp., n holds for A 1,..., A n. Proof. We prove he case for n. The proof for n is similar. Wihou loss of generaliy we may assume ha A 1,..., A n 1 A n. Here Gn A A n A n, L1 A 1 A n and Ln

5 GCD AND LCM-LIKE IDENTITIES FOR IDEALS 5 Ln; A 1,..., A n Ln 1; A 1,..., A n 1. For 2 k n 1, Lk Lk; A 1,..., A n A i1 A ik 1 For n even, A n A i1 A ik 1 i 1 < <i k 1 n 1 Lk; A 1,..., A n 1 Lk 1; A 1,..., A n 1. 1 k n/2 1 Gn 1 k n/2 L2k A i1 A ik 1 L2k; A 1,..., A n 1 L2k 1; A 1,..., A n 1 Ln 1; A 1,..., A n 1 A n On he oher hand, 1 k n/2 1 0 k n/2 1 1 l n 1 Ll; A 1,..., A n 1. L2k + 1 L1; A 1,..., A n L2k + 1; A 1,..., A n 1 L2k; A 1,..., A n 1 A n L1; A 1,..., A n 1 A n 1 l n 1 2 l n 1 Ll; A 1,..., A n 1 Ll; A 1,..., A n 1. Thus he wo sides of n agree. The case where n is odd is similar. Lemma 2.2. Le R be commuaive ring wih ideniy. 1 If A 1,..., A n are ideals R, hen n resp., n holds for A 1,..., A n if and only if n resp., n holds for he localizaions A 1M,..., A nm for each maximal ideal M of R. 2 The condiion n resp., n holds for all ideals of R if and only if n resp., n holds for all ideals of R M for each maximal ideal M of R. 3 If n resp., n holds for all ideals of R and S is a muliplicaively closed subse of R, hen n resp., n holds for all ideals of R S. Proof. This easily follows from he fac ha localizaion preserves sums, producs and inersecions and ha wo ideals I and J of R are equal

6 6 D. D. ANDERSON, S. IZUMI, Y. OHNO, AND M. OZAKI if and only if hey are locally equal, ha is, I M J M for each maximal ideal M of R. Theorem 2.3. Le R be a commuaive ring wih ideniy and le A 1,..., A n be ideals of R. Suppose ha he se {A 1M..., A nm } of localizaions forms a chain for each maximal ideal M of R. Then we have he equaliies n and n for A 1,..., A n. Proof. This follows from Lemmas 2.1, 2.2. The following is he main resul of his secion. Theorem Suppose ha R is an arihmeical ring. Then n and n holds for all ideals A 1,..., A n of R. 2 Suppose ha R is a Prüfer ring. Then n and n hold for all ideals A 1,..., A n of R where a leas n 1 of hem are regular. 3 Suppose ha R is PVMD and A 1,..., A n are nonzero ideals of R. Then Gn Ln L2k G2k L2k + 1, G2k + 1. In paricular, if R is a Prüfer domain, n and n hold for all ideals A 1,..., A n of R. Proof. 1 Suppose ha R is arihmeical. Then for each maximal ideal M of R, R M is a chained ring. So {A 1M,..., A nm } is a chain. By Theorem 2.3, n and n hold for A 1,..., A n. 2 Suppose ha R is a Prüfer ring. Le M be a maximal ideal of R. Le R [M] and [M]R [M] denoe he large quoien ring of R wih respec o M and he exension of M, respecively see [8, p.234]. Since R is a Prüfer ring, R [M], [M]R [M] is a valuaion pair of he oal quoien ring of R [8, Theorem 10.18]. By [8, Exercise 10 c, p. 248], {A 1M,..., A nm } is a chain. Again by Theorem 2.3, n and n hold for A 1,..., A n. 3 For f R[X], A f is he conen of f, i.e. he ideal of R generaed by coefficiens of f. Suppose ha R is PVMD. Pu R{X} R[X] Nv where N v is he muliplicaively closed subse {f R[X] : A f v R} of R[X]. Since polynomial exensions and localizaion preserve sums, producs and inersecions, we

7 have : GCD AND LCM-LIKE IDENTITIES FOR IDEALS 7 Gn Gn; A 1,..., A n Gn; A 1 R{X},..., A n R{X} and L2k R{X} L2k; A 1,..., A n R{X} L2k; A 1 R{X},..., A n R{X} L2k + 1 R{X} L2k + 1; A 1 R{X},..., A n R{X}. Since R{X} is a Prüfer domain even a Bézou domain [7, Theorem 3.7], we have Gn; A 1 R{X},..., A n R{X} L2k; A 1 R{X},..., A n R{X} L2k + 1; A 1 R{X},..., A n R{X}. Bu, for any nonzero ideal A of R, we have AR{X} R A [7, Lemma 3.13]. Hence Gn L2k Gn L2k R{X} R L2k + 1 R{X} R L2k + 1. The corresponding n ideniy is proved in a similar manner. Theorem 2.5. Le R be a commuaive ring wih ideniy. If R saisfies n for all ideals of R for some n 3, hen R saisfies 2,..., n 1 for all ideals of R. Proof. Suppose ha R saisfies n for some n 3: Ln G2k G2k + 1 for all ideals A 1,..., A n of R. Se A n R in he above equaion. Then Ln; A 1,..., A n 1, R A 1 A n 1 R A 1 A n 1

8 8 D. D. ANDERSON, S. IZUMI, Y. OHNO, AND M. OZAKI For 1 k < n, Ln 1; A 1,..., A n 1, Gn; A 1,..., A n 1, R A A n 1 + R R. Gk; A 1,..., A k 1, R 1 A i1 + + A ik A i1 + + A ik Gk 1; A 1,..., A n 1. So, subsiuing A n R in n, we obain n 1 Ln 1 G2k 1 1 Now 2,..., n 1 follow by reversed inducion. G2k + 1. In Example 3.2, we show ha n / n 1 for n 3. Thus here is no analogous resul for n. If in n we se A n 0 or any A i 0, boh sides collapse o 0 while if we se A n R or any A i R, boh sides become 1 k n 1 Lk; A 1,..., A n 1 he analogous formula where A i is deleed. Of course if in n we se A n 0 or any A i 0, boh sides collapse o 0. Theorem 2.6. For a commuaive ring R wih ideniy, he following condiions are equivalen. 1 R is a Prüfer ring. 2 For each n 2, R saisfies n for all ses of n ideals A 1,..., A n where a leas n 1 of hem are regular. 3 For some n 2, R saisfies n for all ses of n regular ideals A 1,..., A n. Proof. 1 2: This is proved in Theorem : Clear. 3 1: Suppose ha R saisfies n for all ses of n regular ideals A 1,..., A n. Then by he proof of Theorem 2.5, R saisfies 2 for all pairs of regular ideals A 1, A 2. Hence R is a Prüfer ring by [3, Theorem 4]. Corollary 2.7. Le R be a commuaive ring wih ideniy and le n 2 be an ineger. Suppose ha R saisfies n for each se A 1,..., A n of regular ideals of R. Then R saisfies m and m for each ineger m 1 and ideals A 1,..., A m wih a leas m 1 of hem regular. Hence if an inegral domain saisfies n for some ineger n 2, i saisfies m and m for all inegers m 1.

9 GCD AND LCM-LIKE IDENTITIES FOR IDEALS 9 Proof. Suppose ha R saisfies n for some paricular n 2 and all regular ideals A 1,..., A n. By Theorem 2.6, R is a Prüfer ring. Then by Theorem 2.4, for each naural numbers m he case m 1 is rivial, R saisfies m and m for all ses of m ideals A 1,..., A m wih a leas m 1 of hem regular. I goes back a leas o Krull ha an inegral domain R is a Prüfer domain if and only if A+BA B AB for all ideals A and B of R. Indeed, if we ake A a and B b o be nonzero principal ideals, we have a, ba b ab; so a, b is inverible, being a facor of a principal ideal. By inducion, each finiely generaed nonzero ideal is inverible; i.e. R is a Prüfer domain. The resul ha R is a Prüfer ring if and only if A + BA B AB for all ideals A and B wih a leas one regular appears o firs be given in [4]. Noe ha [4] uses he erm Prüfer ring for wha we have called an arihmeical ring. We nex prove he formulas GCD n and LCM n for a GCD domain. Theorem 2.8. Le R be a GCD domain and a 1,..., a n n 2 nonzero elemens of R. Then GCD n gcda 1,..., a n lcma i1,..., a i2k and LCM n a 1 a n lcma 1,..., a n a 1 a n 1 i 1 < <i 2k n 2 2k+1 n 1 i 1 < <i 2k+1 n 1 i 1 < <i 2k n 2 2k+1 n 1 i 1 < <i 2k+1 n lcma i1,..., a i2k+1 gcda i1,..., a i2k gcda i1,..., a i2k+1. Proof. Recall ha a GCD domain is a PVMD wih b 1 b n and b 1,..., b n principal for all nonzero b 1,..., b n R. Moreover, if b 1 b n d resp., b 1 b n e, hen d lcmb 1 b n resp., e gcdb 1 b n. Recall ha lcm and gcd are only deermined up o a uni muliple. Throughou he proof below, we will use he fac ha AB AB A B for nonzero ideals A and B of R. Pu A i a i. Noe ha Gn a a n gcda 1,..., a n, Ln a 1 a n a 1 a n lcma 1,..., a n Ln.

10 10 D. D. ANDERSON, S. IZUMI, Y. OHNO, AND M. OZAKI Also, G1 a 1 a n a 1 a n L1. For 1 < k < n, we have Gk a i1,..., a ik a i1,..., a ik and gcda i1,..., a ik Lk lcma i1,..., a ik gcda i1,..., a ik a i1 a ik gcda i1,..., a ik lcma i1,..., a ik. Since each a i1 a ik and hence Lk is principal, we have Lk Lk. Now, since R is a PVMD, so we have Gn L2k L2k + 1 and hence Gn L2k L2k + 1. Since boh sides of he equaliy are principal, we have Gn L2k L2k + 1 as ideals. This proves he firs equaliy up o uni muliplicaion. The proof of he second is similar. Corollary 2.9. Le R be an inegral domain wih he propery ha R M is a GCD-domain for each maximal ideal M of R. Then for each n 2, Gn L2k L2k + 1 holds for all locally principal e.g. inverible ideals A 1,..., A n of R. Proof. Since i suffices o prove he conainmen locally Lemma 2.2, we can assume ha R is a quasi-local GCD domain and ha he ideals A 1,..., A n are principal. If one of he A i 0, boh sides of he conaimen reduce o 0. So we can assume ha A 1,..., A n are nonzero. Now from he proof of Theorem 2.8, we have Gn L2k L2k + 1. Since Gn Gn he resul follows. Example 3.3 shows ha he conainmen in Corollary 2.9 may be sric.

11 GCD AND LCM-LIKE IDENTITIES FOR IDEALS Examples In his secion we give some examples which illusrae he resuls of Secion 2. Example 3.1 n / n for n 3. Le R, M be a quasi-local ring wih M n 0 n 2. We claim ha R saisfies n, bu need no saisfy n. Le A 1,..., A n be ideals of R. If each A i is a proper ideal, hen R saisfies n and n for A 1,..., A n as boh sides reduce o 0. Suppose some A i R, hen n holds as boh sides become Lk; A 1,..., Âi,..., A n 1 k n 1 see he paragraph afer he proof of Theorem 2.5. So, n holds for R. By similar reasoning R saisfies m for m n. Now for n 2, 2 holds and hence so does 2. Suppose n 3 and consider he special case R n k[x, Y ]/X, Y n for a field k. So R n saisfies n for each n 2. However, R n does no saisfy m for any m 2. For, if R n saisfies m for some m 2, hen R n saisfies 2 by Theorem 2.5. This conradics he sric inclusion X, Y X Y X, Y X Y XY. Noe ha for n 3, R n saisfies 3 bu no 2. This is a counerexample of he analogue of Theorem 2.5. We generalize his in he nex example. Example 3.2 n / n 1 for n 3. While n implies n 1 for n 3, we show ha his need no be he case for n. Take R n k[x 1,..., X n ]/X 1,..., X n n for a field k n 3. So R n saisfies n by Example 3.1. In fac R n saisfies m for each m n. However, for 2 m < n, m fails for A 1 X 1,..., A m X m. The proof is essenially he same as he proof ha Gn L2k L2k + 1 in k[x 1, X 2,...] given in Example 3.3. So R n saisfies m precisely for m n. Example 3.3 A ring wih Gn L2k L2k+1 for all n 2. Pu R k[x 1, X 2,...] for a field k and ake A i X i. Noe ha X i1 X ik X i1 X ik. Then Gn X 1,..., X n

12 12 D. D. ANDERSON, S. IZUMI, Y. OHNO, AND M. OZAKI and Li X 1 X n n 1 i 1. I follows ha L2k X 1 X n 2k 1 n 1 X1 X n 2k 1 n 1, n 1 i L2k + 1 Since n 1 i0 and hence Gn L2k X 1 X n n 1 2k X 1 X n n 1 2k. 0, we have L2k L2k + 1. L2k + 1 Noe ha R is a UFD and hence a GCD domain. So Corollary 2.9 gives ha Gn 2k 2k + 1. This example shows ha conainmen may be sric. We previously menioned ha in [2] we show ha for any commuaive ring we have Ln 2 2k n G2k 1 2k+1 n G2k + 1. In [2] we give examples o show ha his inclusion may be sric for all n 2 and ha in general any relaion beween G3L2 and L1L3 is possible. References [1] Anderson, D. D.: GCD domains, Gauss lemma, and conens of polynomials. Non-Noeherian commuaive ring heory, 1-31, Mah. Appl., 520, Kluwer Acad. Publ., Dordrech, [2] Anderson, D. D., Izumi, Shuzo, Ohno, Yasuo, Ozaki, Manabu: A GCD and LCM-like inequaliy for muliplicaive laices, preprin. [3] Anderson, D. D., Pascual, J.: Characerizing Prüfer rings via heir regular ideals. Comm. Algebra , [4] Bus, Huber S., Smih, William: Prüfer rings. Mah. Z [5] Gilmer, Rober: Muliplicaive ideal heory. Correced reprin of he 1972 ediion. Queen s Papers in Pure and Applied Mahemaics, 90. Queen s Universiy, Kingson, ON, [6] Griffin, Malcolm: Some resuls on v-muliplicaion rings. Canad. J. Mah , [7] Kang, B. G.: Prüfer v-muliplicaion domains and he ring R[X] Nv. J. Algebra , [8] Larsen, Max. D., McCarhy, Paul J.: Muliplicaive heory of ideals. Pure and Applied Mahemaics, Vol. 43. Academic Press, New York-London, [9] Wolfram Res.: Funcions, Wolfram. com., hp://funcions.wolfram.com/ [10] Zafrullah, Muhammad: Puing -inveribiliy o use. Non-Noeherian commuaive ring heory, , Mah. Appl., 520, Kluwer Acad. Publ., Dordrech, 2000.

13 GCD AND LCM-LIKE IDENTITIES FOR IDEALS 13 address: Deparmen of Mahemaics, The Universiy of Iowa, Iowa ciy, IA 52242, USA address: Deparmen of Mahemaics, Kindai Universiy, Higashi-Osaka, , Japan address: Mahemaical Insiue, Tohoku Universiy, Sendai, , Japan address: Deparmen of Mahemaics, Waseda Universiy, Shinjuku-ku Tokyo, , Japan