International Journal of Algebra, Vol. 5, 2011, no. 27, 1313 - 1325 On the Structure of Quaternion Rings over Zp C. J. Miguel Instituto de Telecomunica¸c˜oes P´olo de Covilh˜a Beira Interior University Department of Mathematics [email protected] R. Serˆodio Centro de Matem´atica Beira Interior University Department of Mathematics Abstract In this paper we investigate the structure of quaternion rings over the finite field Zp. This work is based on M. Aristidou and A. Demetre’s paper, “A Note on Quaternion Rings over Zp”, who have point out a false theorem of Kandasamy published in “On Finite Quaternion Rings and Skew Fields”. We obtain more detailed descriptions of the struc- ture of Zp[i, j, k], in particular we find the number of zero-divisors of a quaternion ring over Zp and provide a detailed structural description of the zero-divisor graph of a quaternion ring over Zp. Mathematics Subject Classification: 15A03, 15A33, 15A30, 20H25 Keywords: quaternion, ring, division ring, finite field, zero divisor, idem- potent, zero divisor graph 1 Introduction The history of associative algebras begins with Hamilton’s discovery of the real quaternions in 1843. Before 1906 no other skew fields were known besides the quaternions. In that year, however, a completely new kind of skew field was described by Dickson (see [8], for details). The idea that the real quaternions, viewed as an algebra of dimension 4 over the real field with basis {1, i, j, k}, was later generalized by Dickson, replacing the field R by an arbitrary field 1314 C. J. Miguel and R. Serˆodio F of characteristic different from 2 and the elements i and j by two arbitrary elements linearly independent and quadratic over F. This idea is presented in the following definition. Definition 1.1 Let a and b be two non-zero elements of a field F whose characteristic is not 2. Let A be the four dimensional F -vector space with basis {1, i, j, k} and the bilinear multiplication defined by the conditions that 1 is a unity element, and i2 = a, j2 = b, ij = −ji = k. F a,b A is called a quaternion ring over the field and denoted by A = F . The notion of a quaternion ring can be seen as a generalization of the Hamilton quaternions to an arbitrary base field. The Hamilton quaternions −1,−1 R are the case A = R , where is the field of real numbers. By a famous theorem of Frobenius there are only two real quaternion alge- bras: 2× 2 matrices over the reals and Hamilton’s quaternions [5]. In a similar way, over a field F with characteristic different from 2, there are exactly two quaternion algebras: 2 × 2 matrices over F and a division algebra [12]. −1,−1 Hamilton quaternions R form a division algebra. The following ques- tions arise immediately: F a,b 1. For what choices of a, b and the field , is A = F a division algebra? a,b 2. If A = F is not a division algebra what is its structure? For question 1 see [12]. In this paper we explore question 2 for the case where F is the finite field Zp, with p a odd prime and a = b = −1. We use the −1 −1 notation Z [i, j, k] for , . p Zp The paper is organized as follows: in Section 2 we determine the number of zero divisors and idempotents in a quaternion ring over the finite field Zp. Also we construct an isomorphism between the quaternion ring over the finite field Zp and the ring of square matrices of order 2 over the finite field Zp. In section 3 we study the directed and undirected graphs of a quaternion ring over the finite field Zp. 2 Zero Divisors and Idempotents in Zp[i, j, k] In [6] M. Aristidou and A. Demeter show that Zp[i, j, k] does not form a finite division ring as mentioned by Kandasamy [10]. In fact, by the well known Wedderburn’s Little Theorem [11], every finite division ring is a field, that is, commutative. Since, for the finite field Zp, the quaternion ring Zp[i, j, k] Structure of quaternion rings over Zp 1315 is finite and noncommutative, it cannot be a field. Then, Zp[i, j, k] has non trivial zero divisors and non trivial idempotents. In this section we find the number of zero divisors and the number of idempotents of the quaternion ring Zp[i, j, k]. Theorem 2.1 Let p be an odd prime number. Then, the number of zero- divisors in Zp[i, j, k] is p3 + p2 − p. (1) Proof. Since, Zp[i, j, k] is not a division algebra, it is isomorphic to the matrix ring M2(Zp). So, Zp[i, j, k] and M2(Zp) have the same number of zero- divisors. A matrix a b A = ∈ M2(Z ) c d p is singular (a zero-divisor) if and only if the determinant is zero in the field Zp, that is, if and only if ad − bc =0 mod p (2) Thus, the number of zero-divisors in the quaternion ring Zp[i, j, k] is the same as the number of solutions of the congruence (2) in Zp. To find the number of solutions of (2), we will consider the following cases: 1. Suppose first that a =6 0 mod p and fix a, b and c. Then, the equation in the variable x ax = bc mod p, (3) from the theory of linear congruences [9], has a unique solution mod p. There are p2(p − 1) equations of the form (3). Thus, if a =6 0 mod p there are p2(p − 1) singular matrices. 2. Suppose that a = 0 mod p and c =6 0 mod p. Then, the matrix is singular if and only if cb =0 mod p. (4) No restriction takes place for d. There are p − 1 equations of the form (4) each of which has a unique solution. Thus, there are p(p−1) singular matrices of this form. 3. Now suppose that a = 0 mod p and c = 0 mod p. Then the matrix is singular for every b and d. There are p2 of such matrices. Thus, the number os singular matrices in M2(Zp) is p2(p − 1) + p(p − 1) + p2 = p3 + p2 − p. (5) 1316 C. J. Miguel and R. Serˆodio 2 It is well known that if R is a ring, with unity, every ideal in the complete matrix ring Mn(R), n a positive integer, is of the form Mn(I), where I is an ideal of R [11]. This implies that Mn(R) is simple if and only if R is simple. It is clear that a field is necessary simple, thus M(Zp) is simple. A simple non nilpotent artinian ring has only a central idempotent element, namely its unity element, but there may be many non central idempotents elements. In the next theorem we find the number of idempotents elements in Zp[i, j, k]. Theorem 2.2 Let p be an odd prime number. Then, the number of idem- potent elements in Zp[i, j, k] is p2 + p +2. (6) Proof. As we have seen in last result, Zp[i, j, k] ∼= M2(Zp). Thus, the number of idempotent elements in Zp[i, j, k] is exactly the same as in M2(Zp). A matrix A is idempotent if and only if A2 = A mod p, (7) or yet A2 − A =0 mod p, (8) A(A − I)=0 mod p, (9) where I is the identity matrix. Let a b A = ∈ M2(Z ). (10) c d p Then, A is idempotent if and only if a b a − 1 b 0 0 = mod p. (11) c d c d − 1 0 0 Let’s divide this problem into four cases: 1. If b = c =0 mod p, from (11) we have a(a − 1) = 0 mod p, (12) d(d − 1) = 0 mod p. (13) There are only two solutions for a and d. Namely, 0 and 1. Thus, in this case, there are 4 idempotent matrices. Structure of quaternion rings over Zp 1317 2. If b =0 mod p and c =06 mod p, from (11) we have a(a − 1) = 0 mod p, (14) d(d − 1) = 0 mod p, (15) c(a + d − 1) = 0 mod p. (16) From equations (14) and (15), we get that the values for a and d must be 0 or 1. But, from equation (16), since c =6 0 mod p, we must have a + d =1 mod p. The only combinations possible are a = 0 and d = 1 or a = 1 and d = 0. For each of these combinations, c can take p − 1 values. Thus there are 2 × (p − 1) idempotent matrices of this form. 3. If b =6 0 mod p and c = 0 mod p, by symmetry, we have the same number of idempotent matrices as in the last item, i.e, 2 × (p − 1). 4. If b =06 mod p and c =06 mod p, condition (11) implies a2 − a + bc = 0 mod p, (17) b(a + d − 1) = 0 mod p, (18) c(a + d − 1) = 0 mod p, (19) d2 − d + bc = 0 mod p. (20) From equations (18) and (19) we get that a + d = 1 mod p. There are p possible combination, one for each value of Zp. But, from these, we must remove a =0 mod p and a =1 mod p, because these values imply from equation (17) that bc = 0 mod p, which contradicts the initial hypothesis.