On the Structure of Quaternion Rings Over Zp 1 Introduction

Home , Matrix ring, Noncommutative ring, Zero divisor

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¸cões Pólo de Covilhã 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 structure 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, idempotent, 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. Serôdio

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 algebras: 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

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 idempotent 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. Thus we have only p − 2 combinations possible for a and d. Since a + d = 1, from equations (17) and (20) we get

a2 − a + bc = (1 − d)2 − (1 − d)+ bc mod p (21) = 1 − 2d + d2 − 1+ d + bc mod p (22) = d2 − d + bc =0 mod p. (23)

Now, as we have seen, for each of these combinations, a2 − a = d2 − d = −bc mod p. Since bc =06 mod p and (Zp\{0}, ·) is a group, there exist exactly p−1 combinations of b and c for each a and d. Thus, the number of idempotent matrices in this case is (p − 2) × (p − 1).

Summing all these possibilities, we obtain

4+2(p − 1)+2(p − 1)+(p − 2)(p − 1) = p(p +1)+2, (24) 1318 C. J. Miguel and R. Serˆodio from where we conclude that there are p2 + p + 2 idempotent elements in Zp[i, j, k]. 2

Next we ﬁnd explicitly an isomorphism

Φ : Zp[i, j, k] −→ M2(Zp). (25) It is well known that in a ﬁnite ﬁeld F every element is a sum of two squares, that is, the equation

2 2 α = x1 + x2 (26) has, for every α ∈ F, solutions with x1, x2 ∈ F [14]. Then, there exits a, b ∈ Zp such that a2 + b2 = p − 1. (27) Let Φ : Zp[i, j, k] −→ M2(Zp), (28) such that

Φ(x0 + x1i + x2j + x3k)= (29) 1 0 0 1 a b b p − a = x0 + x1 + x2 + x3 (30). 0 1 p − 1 0 b p − a p − a p − b

We will prove that Φ is an isomorphism from Zp[i, j, k] to M2(Zp). First we observe that Φ is a bijection. In fact if we consider Zp[i, j, k] and M2(Zp) as 4-dimensional linear spaces over Zp, Φ is a linear map from Zp[i, j, k] to M2(Zp). Let us to ﬁnd the null space of Φ. Suppose that

0 0 Φ(x0 + x1i + x2j + x3k)= . (31) 0 0 Equation (31) is equivalent to the homogeneous system of linear equations in the variables x0, x1, x2, x3

x0 + ax2 + bx3 =0  x1 + bx2 +(p − a)x3 =0  . (32)  (p − 1)x1 + bx2 +(p − a)x3 =0 x0 +(p − a)x2 +(p − b)x3 =0  The coeﬃcient matrix of the linear system (32) is 1 0 a b  0 1 b p − a  S = . 0 p − 1 b p − a    1 0 p − a p − b    Structure of quaternion rings over Zp 1319

Let us ﬁnd the determinant of the matrix S 1 0 a b 1 0 a b 0 1 b p − a 0 1 b p − a det(S)= = = 0 p − 1 b p − a 0 p − 1 b p − a

1 0 p − a p − b 0 0 p − 2a p − 2b

1 0 a b 0 1 b p − a 2 2 = =2b(p − 2b) − (p − 2a)(2p − 2a)= −4b − 4a = 00 2b 2p − 2a

0 0 p − 2a p − 2b

2 2 = −4(b + a )= −4(p − 1)=4 =06 , since we have supposed that p =6 2. Then Φ is an isomorphism from the linear space Zp[i, j, k] to the linear space M2(Zp). Is is easy to proof that Φ is multiplicative. So Φ is an isomorphism of the ring Zp[i, j, k] to the ring M2(Zp).

It is well known that the multiplicative group of unities of a ﬁnite ﬁeld is a cyclic group [14]. The multiplicative group of Zp[i, j, k] is not abelian, then it can not be cyclic. It is natural to ask what is the minimum of generators for the group of unities of Zp[i, j, k]. The answer is given by the following theorem.

Theorem 2.3 [15] Let F be a ﬁnite ﬁeld. Then the group GL(n, F) of invertible n by n matrices with entries in F can be generated by two elements.

Thus, the multiplicative group of Zp[i, j, k] can be generated by two elements.

3 Zero-divisor Graph of Zp[i, j, k]

The concept of a zero-divisor graph of a commutative ring was first introduced in [3], and later redefined by Anderson and Livingston in [2]. Redmond in [13] extended this concept to the noncommutative case. We recall the definition and some basic properties of a (directed) zero-divisor graph for a noncommutative ring. Let R be a noncommutative ring, the (directed) zero-divisor graph of R, denoted by Γ(R), is the (directed) graph with vertices Z(R)∗ = Z(R) −{0}, the set of non zero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, there is an edge x → y if and only if xy = 0. Note that if x and y are two distinct vertices and xy = yx = 0, then there are two directed edges between x and y and we say there is a multiple edge between x and y. Redmond also defined 1320 C. J. Miguel and R. Serôdio the simple undirected graph, denoted by Γ(R), with vertices in the set Z(R)∗ and such that for two distinct vertices x and y there is an edge connecting them if and only if xy = 0 or yx = 0. Note that, for a commutative ring R, the definition of the zero-divisor graph of R in [2] coincides with the definition of Γ(R). For any vertex a of Γ(R), the number of edges of Γ(R) of the form x → a is called the in-degree of a and the number of edges of Γ(R) of the form a → y is called the out-degree of a. For the (undirected) graph Γ(R) the degree of a vertex is the number of edges incident to the vertex. If there exist vertices x0,...,xn ∈ Γ(R) such that P : x0 → x1 → . . . → xn−1 → xn where xi =6 xj for all i, j =1,...,n (i =6 j) for some positive integer n, then P is called a directed path from x0 to xn of length n. If x0 = xn the path is called a cycle of length n. A directed graph is called connected if there exists a directed path connecting any two distinct vertices. The distance between two distinct vertices x and y, denoted d(x, y), is the length of the shortest directed path connecting then (if such a path does not exit, then d(x, y)= ∞). The diameter of a graph Γ(R), denoted diam(Γ(R)), is equal to

sup{d(x, y) : x, y distinct vertices of Γ(R)} (33)

The girth of a graph Γ(R), denoted g(Γ(R)), is the length of the shortest cycle in Γ(R).

3.1 Vertices, In-degree and Out-degree of Γ(Zp[i,j,k])

Since, Zp[i, j, k] ∼= M2(Zp) obviously Γ(Zp[i, j, k]) =∼ Γ(M2(Zp)). Thus, by 3 2 Theorem 2.1, Γ(Zp[i, j, k]) has p + p − p − 1 vertices. In the following theorem, we will determine the in-degree and the out- degree of each vertice of Γ(Zp[i, j, k]).

Theorem 3.1 The in-degree and the out-degree of a vertice A of a zero- 2 2 2 2 divisor graph Γ(M2(Zp)) are p − 1 if A =06 and p − 2 if A =0.

Proof. (see [1] for a diﬀerent proof.) Observe that, if A, X ∈ M2(Zp) the matrix equation (in the variable X)

AX =0, (34) is equivalent to AX1 = AX2 = 0, where X1 and X2 are the ﬁrst and the second columns of the matrix A, respectively. The set of solutions of a homogeneous linear system MY = 0 is a subespace of dimension m − k, where m is the number of columns of the matrix M and k is the characteristic of the matrix M. Therefore, for each i = 1, 2, the set of solutions of AXi = 0 forms a Structure of quaternion rings over Zp 1321

subpace of dimension 1. Thus, there are exactly p solutions for each AXi = 0, i =1, 2. Then, equation (34) has p2 − 1 non zero solutions. Equation (34) is equivalent to XT AT = 0, and thus the right annihilator of A has the same cardinality has the left annihilator of AT . Since, the characteristic of A is the same has the characteristic of AT , we conclude that the right and the left annihilators of A have the same cardinality. In the case that A2 = 0, since a zero-divisor graph has no loops, the in- degree and the out-degree of the vertex A are p2 − 2. If A2 =6 0 the in-degree and the out-degree of the vertex A are p2 − 1.2

3.2 The graph Γ(Zp[i,j,k])

It is clearly that Γ(Zp[i, j, k]) and Γ(Zp[i, j, k]) have the same vertices. To ﬁnd the degree of each vertex of Γ(Zp[i, j, k]) we need a little lemma.

Lemma 3.2 Let A ∈ M2(Zp) be a singular matrix. Then, the equation

AX = XA =0, (35) has p solutions in M2(Zp).

Proof. Let Zp be the algebraic closure of the finite field Zp. Since in an algebraic closed field we can reduce a matrix to the Jordan Canonical Form, −1 there exists a regular matrix S ∈ M2(Zp) such that SAS = JA, where JA is the jordan canonical form of the matrix A. Since, rank(A) = 1 we can assume (for some non zero α ∈ Zp)

α 0 0 1 JA = or JA = . 0 0 0 0 −1 The function Ψ : M2(Zp) → M2(Zp) deﬁned by Ψ(L) = SLS is a ring monomorphism from M2(Zp) to M2(Zp). Therefore, Ψ (M2(Zp)) is a subring of M2(Zp) isomorphic to M2(Zp). So, the number of solutions of equation (35) in M2(Zp) is the same has the number of solutions of

SAS−1SXS−1 = SXS−1SAS−1 =0, (36) −1 −1 in Ψ(M2(Zp)). The solutions of equation SAS SXS = 0 are of the form

−1 0 0 SXS = , z w and the solutions of SXS−1SAS−1 = 0 are of the form 1322 C. J. Miguel and R. Serˆodio

−1 0 y SXS = . 0 w Thus, the solutions of equation (36) are the p matrices of the form

−1 0 0 SXS = . 0 w 2 2 Let A ∈ Γ(M2(Zp) be a vertex such that A =6 0. Since Γ(M2(Zp) has no multiples edges, applying theorem 3.1 and lemma 3.2 we conclude that the 2 degree of the vertex A is 2p − p − 1. If A ∈ Γ(M2(Zp) is a vertex such that A2 = 0 in a similar manner we can conclude that the degree of the vertex A is 2p2 − p − 2. Thus, we have the following

Theorem 3.3 The degree of a vertice A of a zero-divisor graph Γ(M2(Zp)) is 2p2 − p − 1 if A2 =06 and 2p2 − p − 2 if A2 =0.

In [7] Fine and Herstein determine the number of n by n nilpotent matrices over a finite field of characteristic p. following Fine and Herstein results the 2 number of non zero nilpotent matrices in M2(Zp) is p − 1. Observing that a non zero nilpotent matrix in M2(Zp) necessarily as index of nilpotence 2, we 3 2 conclude that the graph Γ(Zp[i, j, k]) has p −p vertices with degree 2p −p−1 and p2 − 1 vertices with degree 2p2 − p − 2. In figure 1 we present the graph Γ(Z3[i, j, k]). This graph has 32 vertices, 24 vertices of degree 14 and 8 vertices of degree 13.

3.3 Diameter of Γ(Zp[i,j,k]) and Girth of Γ(Zp[i,j,k]) In [13], is proved that for any ring R the graph Γ(R) is connected if every left zero divisor is also a right zero divisor. Since, for every ﬁeld F and any positive integer n, the right and the left zero divisors of the matrix ring Mn(F) coincide (in fact, there are the matrices with zero determinant), we conclude that the graph Γ(Zp[i, j, k]) is connected. Since, for two vertices A, B ∈ Γ(M2(Zp)) there exist regular matrices P, Q ∈ M2(Zp) such that

a 0 0 0 AP = and QB = , (37) b 0 c d b b b

Structure of quaternion ringsb over Zp b 1323

b b

b b b

Figure 1. Γ(Z3[i, j, k]) if we put 0 0 C = P Q, (38) 1 0 we have C =6 0 , AC = 0 , CB = 0. Thus, we have

Theorem 3.4 Diam(Γ(Zp[i, j, k])) = 2.

If we put 1 1 1 p − 1 0 1 A = , B = and C = , 0 0 p − 1 1 0 1 A, B and C are distinct non zero matrices and AB = BC = CA = 0. Thus, we conclude

Theorem 3.5 The girt of Γ(Zp[i, j, k]) is 3, that is, g(Γ(Zp[i, j, k])) = 3. 1324 C. J. Miguel and R. Serˆodio

3.4 Planarity of Γ(Zp[i,j,k]) A planar graph is a graph that can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. We will use Kuratowski’s Theorem, which states that a graph is planar if and only if it does not contained a subdivision of the complete graph K5 or of the complete bipartite graph K3,3 [4]. Clearly the directed zero-divisor graph Γ(R) is planar if and only if the undirected zero-divisor graph Γ(R) is planar.

Theorem 3.6 The zero-divisor graph Γ(Zp[i, j, k]) is not planar.

Proof. Since, in the graph Γ(M2(Zp)) the vertices

0 0 0 0 0 0 , , (39) 0 1 1 1 2 1 are all adjacent to

1 0 0 0 1 0 , , (40) 0 0 1 0 1 0

the bipartite complete graph K3,3 is a subgraph of Γ(M2(Zp)). So, by Kuratowski’s Theorem, Γ(M2(Zp)) is not planar. 2

References

[1] S. Akbari, A. Mohammadian, Zero-divisor graphs of non-commutative rings, Journal of Algebra, 296 (2006), 462-479.

[2] D. F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring, Journal of Algebra, 217 (1999), 434-447.

[3] I. Beck, Coloring of commutative rings, Journal of Algebra, 116 (1988), 208-226.

[4] Bla Bollobs, Modern Graph Theory, Grad. Texts in Math., Vol. 184, Springer-Verlag, New York, 1998.

[5] H. -D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirche, A. Prestel, R. Remmert, Numbers, Springer-Verlag, New York, 1991.

[6] M. Aristidou and A. Demetre, A Note on Quaternion Rings over Zp, International Journal of Algebra, Vol.3 (2009), no. 15, 725–728. Structure of quaternion rings over Zp 1325

[7] N. J. Fine and I. N. Herstein, The Probability that a Matrix be Nilpotent, Illinois Journal of Mathematics 2 (1958), 499–504.

[8] J. J. Gray and K. H. Parsshall, Episodes in the History of Modern Al- gebra (1800-1950), American Mathematical Society and London: London Mathematical Society, 2007.

[9] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1971.

[10] W. B. V. Kandasamy, On Finite Quaternion Rings and Skew Fields, Acta Ciencia Indica, Vol. XXVI, No 2. (2000), 133–135.

[11] T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, 1991.

[12] R. S. Pierce, Associative Algebras, Springer, 1982.

[13] S. Redmond, The zero-divisor graph of a noncommutative ring, Interna- tional Journal of Commutative Rings 1(4) (2002), 203-211.

[14] C. Small, Arithmetic of Finite Fields, A Series of Monographs and Text- books, no. 148, Pure and Applied Mathematics, 1991.

[15] W.C. Waterhouse, Two Generators for the General Linear Groups over Finite Fields, Linear and Multilinear Algebra, Vol. 24 (1989), 227–230.

Received: April, 2011