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Menonʼs identity in residually finite Dedekind domains
Journal of Number Theory 137 (2014) 179–185
Contents lists available at ScienceDirect
Journal of Number Theory www.elsevier.com/locate/jnt
Menon’s identity in residually finite Dedekind domains C. Miguel Instituto de Telecomunicações, Beira Interior University, Department of Mathematics, Covilhã, Portugal
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 26 November 2013 Accepted 29 November 2013 Available online 4 January 2014 Communicated by David Goss
In this note we give an extension of the well-known Menon’s identity to residually finite Dedekind domains. © 2013 Elsevier Inc. All rights reserved.
MSC: 11A25 20D99 Keywords: Burnside’s lemma Dedekind domain Group action Residually finite ring
1. Introduction The starting point for this paper is an unusual relation between the divisor function and the Euler totient function, which states that for every n ∈ N = {1, 2, . . .} gcd(a − 1, n) = ϕ(n)τ (n), (1) a∈U (Zn )
where U (Zn ) = {a ∈ Zn : gcd(n, a) = 1}, ϕ is the Euler totient function and τ (n) is the number of positive divisors of n. This interesting arithmetical identity, known as E-mail address: [email protected]. 0022-314X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jnt.2013.11.003
180
C. Miguel / Journal of Number Theory 137 (2014) 179–185
Menon’s identity, is due to P.K. Menon [12]. This identity has been generalized by many authors (see e.g. [5–7,16–18]). However, all these generalizations are in the setting of the traditional domain of the rational integers. As is well-known there are other number systems that are in many ways analogous to the rational integers. For instance, the ring F[X] of all polynomials in one variable with coefficients in a field F, the p-adic integers, the ring OK of integers in a number field K or more generally a Dedekind domain. The question naturally arises as to whether we can establish similar assertions in a more general framework. The aim of this paper is to extend Menon’s identity to a special case of Dedekind domains, namely, residually finite Dedekind domains. That is, Dedekind domains D such that for each non-zero ideal n of D, the residue class ring D/n is finite. The positive rational integer N (n) defined by N (n) = |D/n| is called the norm of the ideal n. Residually finite rings (also called rings of finite norm property) have historically commanded strong interest (see e.g. [8,11]). The reason for this historical interest comes from algebraic number theory. Indeed, the ring OK of integers in a number field (or more generally, in a global field) is a residually finite Dedekind domain. As is well-known factorization into irreducible elements frequently fails for Dedekind domains. But using ideals in place of elements we can save unique factorization. Unique factorization of ideals in a Dedekind domain permits calculations that are analogous to some familiar manipulations involving ordinary integers (for details, see [13, Chapter 1]). Moreover, we can define a generalized Euler totient function type for a non-zero ideal of a Dedekind domain. Let n be a non-zero ideal in a Dedekind domain D, then the generalized Euler totient function, which is denoted by ϕD (n), is defined to be the order of the multiplicative group of units in the factor ring D/n, with the convention that ϕD (D) = 1. That is, ϕD (n) =
1 if n = D, |U (D/n)| otherwise.
Notice that, since we only consider residually finite Dedekind domains, it follows that ϕD is a finite valued function. This function shares the same basic properties as the usual Euler’s totient function. For example
ϕD (n) = N (n)
p|n
1 1− , N (p)
C. Miguel / Journal of Number Theory 137 (2014) 179–185
181
where the product ranges over all prime ideals dividing n [13, p. 13]. Also for a non-zero ideal n we define the generalized divisor function τD (n) to be the number of ideals that divide the ideal n. That is, τD (n) = 1. d|n
Recall that, in a Dedekind domain the ideal d divides the ideal n if and only if the ideal d contains the ideal n. Since a non-zero ideal in a Dedekind domain is contained only in finitely many distinct ideals [13, p. 8], it follows that the function τD is finite valued. Our main result is the following theorem. Theorem 1.1. Let D be a residually finite Dedekind domain, n a non-zero ideal of D and U (D/n) the group of units of D/n. Then, N a − 1 + n = ϕD (n)τD (n). (2) a∈U (D/n)
Formula (1) is a particular case of formula (2). Indeed, every natural number n generates a principal ideal n = {an: a ∈ Z} in the rational integers, and conversely, every non-zero ideal I in the rational integers is generated by a unique natural number n, namely the norm N (I) = |Z/I| of that ideal. Now, for two natural numbers a and b we have that a + b = c, where c = gcd(a, b). Hence, in the case of rational integers we have
N a − 1 + n = N gcd(a − 1, n) = gcd(a − 1, n). 2. Preliminaries In this section we assemble the tools that we require to prove Theorem 1.1. One of the key tools is Burnside’s lemma, also called Cauchy–Frobenius lemma, concerning group actions, see [14]. Burnside’s lemma. Let G be a finite group acting on a finite set X and, for each g ∈ G, let X g = {x ∈ X | g.x = x}, be the set of elements in X that are fixed by g. Then, the number N of distinct orbits is the average number of fixed points of the elements of the group. That is, N=
1 g X . |G| g∈G
This simple lemma is often used in the literature to prove Menon’s-type identities (see e.g. [16,17]). However, the application of Burnside’s lemma in the setting of Dedekind domains is not immediate. We need some results about commutative rings.
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C. Miguel / Journal of Number Theory 137 (2014) 179–185
We begin with the question of the uniqueness of the generators of principal ideals. Let a and b be elements of a commutative ring with identity. The elements a, b are called associated if a = ub for some unit u. Clearly associated elements generate the same principal ideal. The converse is not true as shown by the following example due to Kaplansky [9, p. 466]. Take the set of all pairs (n, f (x)), where n is a rational integer, f (x) a polynomial with coefficients in the Galois field GF (5), and the constant term of f is congruent to n mod 5. Addition and multiplication are componentwise. Then, (0, x) and (0, 2x) generate the same principal ideal but are not associated. Fore more details about this question the reader is referred to [15]. We merely state the following result which will be used later in the paper. Lemma 2.1. Let R be a commutative artinian ring with identity and let a, b ∈ R. Then, a = b if and only if there exists a unit u ∈ R such that a = ub. Proof. By the well-known structure theorem for artinian rings [2, p. 90], R decomposes uniquely (up to isomorphism) as a direct sum of finitely many artinian local rings. That is, R ∼ = R1 ⊕ · · · ⊕ Rk , where each Ri , for i = 1, . . . , k, is a local ring. Therefore, each element r ∈ R can be identified with a k-tuple (r1 , . . . , rk ), where ri ∈ Ri for i = 1, . . . , k, with componentwise addition and multiplication. Hence, the ideal r decomposes as ri ⊕ · · · ⊕ rk . Thus, it suffices to prove the result for the case where R is a local ring. So, let us assume that R is a local ring. The if part is obvious. To prove the only if part, let us assume that a = b. Then, there exist x, y ∈ R such that a = bx and b = ay. If a = 0 or b = 0 there is nothing to prove. So, assume that both a and b are non-zero. If x or y is a unit, then we get the result. Otherwise, since the ring is local it follows that 1 − xy is a unit. Now, a(1 − xy) = 0 and thus a = 0, a contradiction. 2 Recall that if a is an element of a commutative ring R, then the annihilator of a in R is ann R (a) = {a ∈ R: ax = 0}. Note that annR (a) is an ideal of R. Lemma 2.2. Let R be a residually finite commutative ring with identity, I an ideal of R and a ∈ R. If ψ : R → R/I denotes the canonical epimorphism, then, ann R/I ψ(a) = R/ a + I . Proof. Consider the composition of canonical epimorphisms
R −→ R/I −→(R/I)/ ψ(a) . ψ
φ
It is easy to check that the kernel of this composition is a + I. Hence, by the first isomorphism theorem for rings, we obtain
R/ a + I ∼ = (R/I)/ ψ(a) .
(3)
C. Miguel / Journal of Number Theory 137 (2014) 179–185
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Now consider the factor ring R/I as a Z-module and define the Z-linear map L : R/I → R/I by L(x) = ψ(a)x. Since L(R/I) = ψ(a) and ker(L) = ann R/I ψ(a), by the first isomorphism theorem for modules it follows that
(R/I)/ann R/I ψ(a) ∼ = ψ(a) .
(4)
Finally, combining Eq. (3) with Eq. (4) we find that
ann R/I ψ(a) = (R/I)/ ψ(a) = R/ a + I , as claimed. 2 We know from elementary number theory that the linear congruence ax ≡ b (mod n) is solvable if and only if d = gcd(a, n) is a divisor of b. Moreover, if the congruence is solvable it has exactly d pairwise incongruent solutions modulo n [4, p. 62]. In the following theorem we generalize this result to residually finite Dedekind domains. Recall that, two elements a and b of a Dedekind domain D are congruent modulo the ideal n, and write a ≡ b mod n, if the difference a − b belongs to n. Clearly, congruences mod n are equivalent to equalities in the quotient ring D/n. Theorem 2.3. Let D be a residually finite Dedekind domain and n an ideal of D. For a, b ∈ D, the linear congruence ax ≡ b
mod n
(5)
is solvable if and only if b ∈ a + n. Furthermore, if the congruence is solvable, then it has exactly N (a + n) incongruent solutions modulo n. Proof. Suppose that congruence (5) is solvable and let x0 be a solution. So, ax0 − b ∈ n, and therefore b = ax0 + z, for some z ∈ n. Hence, b ∈ a + n. Conversely, if b ∈ a + n, then b = au + v, for some u ∈ D and v ∈ n. Therefore, u is a solution of congruence (5). In order to count the number of solutions we note that congruence (5) is equivalent to the following equation in the quotient ring D/n ψ(a)x = ψ(b),
(6)
where ψ denotes the canonical epimorphism from D to D/n. Note that all solutions of Eq. (6) can be written in the form s + h, where s is a particular solution and h belongs to the annihilator of ψ(a). Hence, the cardinality of the solution set of congruence (5) equals the cardinality of the annihilator of ψ(a) in the factor ring D/n. The result then follows immediately from Lemma 2.2. 2 We end this section by quoting the following result, which is well-known to experts in algebraic number theory. For a proof we refer the reader to [1, p. 8].
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C. Miguel / Journal of Number Theory 137 (2014) 179–185
Lemma 2.4. Let D be a Dedekind domain. For every non-zero ideal n of D and 0 = a ∈ a, there exists b ∈ n such that n = a, b. 3. Proof of the main theorem We can now prove our main result. Proof of Theorem 1.1. Let D be a residually finite Dedekind domain and let n be a non-zero ideal of D. Let the group G = U (D/n) of units in D/n act on X = D/n by (a, b) → a.b. Now, for each a ∈ G the set X a = {b ∈ D/n: ab = b} of elements in X that are fixed by a has order N (a − 1 + n). This is a consequence of Theorem 2.3. To count the number N of orbits of your action, we observe that according to Lemma 2.1 two elements belong to the same orbit if and only if they generate the same ideal. Therefore, the number N of orbits equals the number of principal ideals of D/n. Let us show that D/n is a principal ideal ring. Note that the ideals of D/n are of the form a/n where a is an ideal of D that contains n. So, let a/n be an ideal of D/n. Since n ⊂ a it follows by Lemma 2.4 that there exist b, c, d ∈ D such that a = d, b and n = d, c. Hence, a/n is the principal ideal generated by b + n. Since τD (n) is the number of ideals in D/n it follows that τD (n) equals the number of orbits. Finally, applying Burnside’s lemma we get
N a − 1 + n = ϕD (n)τD (n),
a∈U (D/n)
as required. 2 Remark. Krull domains, first studied by W. Krull in [10], are the most natural class of rings in which there is a divisor theory. Krull domains are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1. It would be interesting to study Menon’s-type identities in these rings. Perhaps one obstacle may be the fact that in a Krull domain there is not a fixed finite bound on the number of elements required to generate an ideal. In fact, it was proved by I.S. Cohen in [3] that if there exists a finite bound on the number of elements required to generate an ideal, then the ring is Noetherian and of Krull dimension one, that is, a Dedekind domain. References [1] Robert B. Ash, A Course in Algebraic Number Theory, Dover Publications, Inc., Mineola, NY, 2010. [2] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley Publishing Co., Reading, MA, London, Don Mills, ON, 1969. [3] I.S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950) 27–42.
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[4] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, sixth edition, Oxford University Press, Oxford, 2008. Revised by D.R. Heath-Brown and J.H. Silverman. With a foreword by Andrew Wiles. [5] Pentti Haukkanen, Menon’s identity with respect to a generalized divisibility relation, Aequationes Math. 70 (3) (2005) 240–246. [6] P. Haukkanen, J. Wang, A generalization of Menon’s identity with respect to a set of polynomials, Port. Math. 53 (3) (1996) 331–337. [7] Pentti Haukkanen, Jun Wang, High degree analogs of Menon’s identity, Indian J. Math. 39 (1) (1997) 37–42. [8] Yasuyuki Hirano, On residually finite rings, Math. J. Okayama Univ. 35 (1993) 155–167. [9] Irving Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949) 464–491. [10] W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1931) 160–196. [11] Kathleen B. Levitz, Joe L. Mott, Rings with finite norm property, Canad. J. Math. 24 (1972) 557–565. [12] P.K. Menon, On the sum Σ(a − 1, n)[(a, n) = 1], J. Indian Math. Soc. 29 (1965) 155–163. [13] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, third edition, Springer Monogr. Math., Springer-Verlag, Berlin, 2004. [14] P.M. Neumann, A lemma that is not Burnside’s, Math. Sci. 4 (1979) 133–141. [15] Dennis Spellman, Georgia M. Benkart, Anthony M. Gaglione, W. David Joyner, Mark E. Kidwell, Mark D. Meyerson, William P. Wardlaw, Principal ideals and associate rings, JP J. Algebra Number Theory Appl. 2 (2) (2002) 181–193. arnˇ auceanu, A generalization of Menon’s identity, J. Number Theory 132 (11) (2012) [16] Marius Tˇ 2568–2573. [17] L. Tóth, Menon’s identity and arithmetical sums representing functions of several variables, Rend. Semin. Mat. Univ. Politec. Torino 69 (1) (2011) 97–110. [18] S. Venkatramaiah, On generalisations of Menon’s identity, Math. Ed. (Siwan) 31 (4) (1997) 220–225.
Contents lists available at ScienceDirect
Journal of Number Theory www.elsevier.com/locate/jnt
Menon’s identity in residually finite Dedekind domains C. Miguel Instituto de Telecomunicações, Beira Interior University, Department of Mathematics, Covilhã, Portugal
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 26 November 2013 Accepted 29 November 2013 Available online 4 January 2014 Communicated by David Goss
In this note we give an extension of the well-known Menon’s identity to residually finite Dedekind domains. © 2013 Elsevier Inc. All rights reserved.
MSC: 11A25 20D99 Keywords: Burnside’s lemma Dedekind domain Group action Residually finite ring
1. Introduction The starting point for this paper is an unusual relation between the divisor function and the Euler totient function, which states that for every n ∈ N = {1, 2, . . .} gcd(a − 1, n) = ϕ(n)τ (n), (1) a∈U (Zn )
where U (Zn ) = {a ∈ Zn : gcd(n, a) = 1}, ϕ is the Euler totient function and τ (n) is the number of positive divisors of n. This interesting arithmetical identity, known as E-mail address: [email protected]. 0022-314X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jnt.2013.11.003
180
C. Miguel / Journal of Number Theory 137 (2014) 179–185
Menon’s identity, is due to P.K. Menon [12]. This identity has been generalized by many authors (see e.g. [5–7,16–18]). However, all these generalizations are in the setting of the traditional domain of the rational integers. As is well-known there are other number systems that are in many ways analogous to the rational integers. For instance, the ring F[X] of all polynomials in one variable with coefficients in a field F, the p-adic integers, the ring OK of integers in a number field K or more generally a Dedekind domain. The question naturally arises as to whether we can establish similar assertions in a more general framework. The aim of this paper is to extend Menon’s identity to a special case of Dedekind domains, namely, residually finite Dedekind domains. That is, Dedekind domains D such that for each non-zero ideal n of D, the residue class ring D/n is finite. The positive rational integer N (n) defined by N (n) = |D/n| is called the norm of the ideal n. Residually finite rings (also called rings of finite norm property) have historically commanded strong interest (see e.g. [8,11]). The reason for this historical interest comes from algebraic number theory. Indeed, the ring OK of integers in a number field (or more generally, in a global field) is a residually finite Dedekind domain. As is well-known factorization into irreducible elements frequently fails for Dedekind domains. But using ideals in place of elements we can save unique factorization. Unique factorization of ideals in a Dedekind domain permits calculations that are analogous to some familiar manipulations involving ordinary integers (for details, see [13, Chapter 1]). Moreover, we can define a generalized Euler totient function type for a non-zero ideal of a Dedekind domain. Let n be a non-zero ideal in a Dedekind domain D, then the generalized Euler totient function, which is denoted by ϕD (n), is defined to be the order of the multiplicative group of units in the factor ring D/n, with the convention that ϕD (D) = 1. That is, ϕD (n) =
1 if n = D, |U (D/n)| otherwise.
Notice that, since we only consider residually finite Dedekind domains, it follows that ϕD is a finite valued function. This function shares the same basic properties as the usual Euler’s totient function. For example
ϕD (n) = N (n)
p|n
1 1− , N (p)
C. Miguel / Journal of Number Theory 137 (2014) 179–185
181
where the product ranges over all prime ideals dividing n [13, p. 13]. Also for a non-zero ideal n we define the generalized divisor function τD (n) to be the number of ideals that divide the ideal n. That is, τD (n) = 1. d|n
Recall that, in a Dedekind domain the ideal d divides the ideal n if and only if the ideal d contains the ideal n. Since a non-zero ideal in a Dedekind domain is contained only in finitely many distinct ideals [13, p. 8], it follows that the function τD is finite valued. Our main result is the following theorem. Theorem 1.1. Let D be a residually finite Dedekind domain, n a non-zero ideal of D and U (D/n) the group of units of D/n. Then, N a − 1 + n = ϕD (n)τD (n). (2) a∈U (D/n)
Formula (1) is a particular case of formula (2). Indeed, every natural number n generates a principal ideal n = {an: a ∈ Z} in the rational integers, and conversely, every non-zero ideal I in the rational integers is generated by a unique natural number n, namely the norm N (I) = |Z/I| of that ideal. Now, for two natural numbers a and b we have that a + b = c, where c = gcd(a, b). Hence, in the case of rational integers we have
N a − 1 + n = N gcd(a − 1, n) = gcd(a − 1, n). 2. Preliminaries In this section we assemble the tools that we require to prove Theorem 1.1. One of the key tools is Burnside’s lemma, also called Cauchy–Frobenius lemma, concerning group actions, see [14]. Burnside’s lemma. Let G be a finite group acting on a finite set X and, for each g ∈ G, let X g = {x ∈ X | g.x = x}, be the set of elements in X that are fixed by g. Then, the number N of distinct orbits is the average number of fixed points of the elements of the group. That is, N=
1 g X . |G| g∈G
This simple lemma is often used in the literature to prove Menon’s-type identities (see e.g. [16,17]). However, the application of Burnside’s lemma in the setting of Dedekind domains is not immediate. We need some results about commutative rings.
182
C. Miguel / Journal of Number Theory 137 (2014) 179–185
We begin with the question of the uniqueness of the generators of principal ideals. Let a and b be elements of a commutative ring with identity. The elements a, b are called associated if a = ub for some unit u. Clearly associated elements generate the same principal ideal. The converse is not true as shown by the following example due to Kaplansky [9, p. 466]. Take the set of all pairs (n, f (x)), where n is a rational integer, f (x) a polynomial with coefficients in the Galois field GF (5), and the constant term of f is congruent to n mod 5. Addition and multiplication are componentwise. Then, (0, x) and (0, 2x) generate the same principal ideal but are not associated. Fore more details about this question the reader is referred to [15]. We merely state the following result which will be used later in the paper. Lemma 2.1. Let R be a commutative artinian ring with identity and let a, b ∈ R. Then, a = b if and only if there exists a unit u ∈ R such that a = ub. Proof. By the well-known structure theorem for artinian rings [2, p. 90], R decomposes uniquely (up to isomorphism) as a direct sum of finitely many artinian local rings. That is, R ∼ = R1 ⊕ · · · ⊕ Rk , where each Ri , for i = 1, . . . , k, is a local ring. Therefore, each element r ∈ R can be identified with a k-tuple (r1 , . . . , rk ), where ri ∈ Ri for i = 1, . . . , k, with componentwise addition and multiplication. Hence, the ideal r decomposes as ri ⊕ · · · ⊕ rk . Thus, it suffices to prove the result for the case where R is a local ring. So, let us assume that R is a local ring. The if part is obvious. To prove the only if part, let us assume that a = b. Then, there exist x, y ∈ R such that a = bx and b = ay. If a = 0 or b = 0 there is nothing to prove. So, assume that both a and b are non-zero. If x or y is a unit, then we get the result. Otherwise, since the ring is local it follows that 1 − xy is a unit. Now, a(1 − xy) = 0 and thus a = 0, a contradiction. 2 Recall that if a is an element of a commutative ring R, then the annihilator of a in R is ann R (a) = {a ∈ R: ax = 0}. Note that annR (a) is an ideal of R. Lemma 2.2. Let R be a residually finite commutative ring with identity, I an ideal of R and a ∈ R. If ψ : R → R/I denotes the canonical epimorphism, then, ann R/I ψ(a) = R/ a + I . Proof. Consider the composition of canonical epimorphisms
R −→ R/I −→(R/I)/ ψ(a) . ψ
φ
It is easy to check that the kernel of this composition is a + I. Hence, by the first isomorphism theorem for rings, we obtain
R/ a + I ∼ = (R/I)/ ψ(a) .
(3)
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Now consider the factor ring R/I as a Z-module and define the Z-linear map L : R/I → R/I by L(x) = ψ(a)x. Since L(R/I) = ψ(a) and ker(L) = ann R/I ψ(a), by the first isomorphism theorem for modules it follows that
(R/I)/ann R/I ψ(a) ∼ = ψ(a) .
(4)
Finally, combining Eq. (3) with Eq. (4) we find that
ann R/I ψ(a) = (R/I)/ ψ(a) = R/ a + I , as claimed. 2 We know from elementary number theory that the linear congruence ax ≡ b (mod n) is solvable if and only if d = gcd(a, n) is a divisor of b. Moreover, if the congruence is solvable it has exactly d pairwise incongruent solutions modulo n [4, p. 62]. In the following theorem we generalize this result to residually finite Dedekind domains. Recall that, two elements a and b of a Dedekind domain D are congruent modulo the ideal n, and write a ≡ b mod n, if the difference a − b belongs to n. Clearly, congruences mod n are equivalent to equalities in the quotient ring D/n. Theorem 2.3. Let D be a residually finite Dedekind domain and n an ideal of D. For a, b ∈ D, the linear congruence ax ≡ b
mod n
(5)
is solvable if and only if b ∈ a + n. Furthermore, if the congruence is solvable, then it has exactly N (a + n) incongruent solutions modulo n. Proof. Suppose that congruence (5) is solvable and let x0 be a solution. So, ax0 − b ∈ n, and therefore b = ax0 + z, for some z ∈ n. Hence, b ∈ a + n. Conversely, if b ∈ a + n, then b = au + v, for some u ∈ D and v ∈ n. Therefore, u is a solution of congruence (5). In order to count the number of solutions we note that congruence (5) is equivalent to the following equation in the quotient ring D/n ψ(a)x = ψ(b),
(6)
where ψ denotes the canonical epimorphism from D to D/n. Note that all solutions of Eq. (6) can be written in the form s + h, where s is a particular solution and h belongs to the annihilator of ψ(a). Hence, the cardinality of the solution set of congruence (5) equals the cardinality of the annihilator of ψ(a) in the factor ring D/n. The result then follows immediately from Lemma 2.2. 2 We end this section by quoting the following result, which is well-known to experts in algebraic number theory. For a proof we refer the reader to [1, p. 8].
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C. Miguel / Journal of Number Theory 137 (2014) 179–185
Lemma 2.4. Let D be a Dedekind domain. For every non-zero ideal n of D and 0 = a ∈ a, there exists b ∈ n such that n = a, b. 3. Proof of the main theorem We can now prove our main result. Proof of Theorem 1.1. Let D be a residually finite Dedekind domain and let n be a non-zero ideal of D. Let the group G = U (D/n) of units in D/n act on X = D/n by (a, b) → a.b. Now, for each a ∈ G the set X a = {b ∈ D/n: ab = b} of elements in X that are fixed by a has order N (a − 1 + n). This is a consequence of Theorem 2.3. To count the number N of orbits of your action, we observe that according to Lemma 2.1 two elements belong to the same orbit if and only if they generate the same ideal. Therefore, the number N of orbits equals the number of principal ideals of D/n. Let us show that D/n is a principal ideal ring. Note that the ideals of D/n are of the form a/n where a is an ideal of D that contains n. So, let a/n be an ideal of D/n. Since n ⊂ a it follows by Lemma 2.4 that there exist b, c, d ∈ D such that a = d, b and n = d, c. Hence, a/n is the principal ideal generated by b + n. Since τD (n) is the number of ideals in D/n it follows that τD (n) equals the number of orbits. Finally, applying Burnside’s lemma we get
N a − 1 + n = ϕD (n)τD (n),
a∈U (D/n)
as required. 2 Remark. Krull domains, first studied by W. Krull in [10], are the most natural class of rings in which there is a divisor theory. Krull domains are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1. It would be interesting to study Menon’s-type identities in these rings. Perhaps one obstacle may be the fact that in a Krull domain there is not a fixed finite bound on the number of elements required to generate an ideal. In fact, it was proved by I.S. Cohen in [3] that if there exists a finite bound on the number of elements required to generate an ideal, then the ring is Noetherian and of Krull dimension one, that is, a Dedekind domain. References [1] Robert B. Ash, A Course in Algebraic Number Theory, Dover Publications, Inc., Mineola, NY, 2010. [2] M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley Publishing Co., Reading, MA, London, Don Mills, ON, 1969. [3] I.S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J. 17 (1950) 27–42.
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