#### Abstract

Let be a Galois ring of characteristic and cardinality , where and are positive integers and is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over is obtained.

#### 1. Introduction and Preliminaries

Let be a prime number, and be positive integers, and a Galois ring of characteristic and cardinality . Then is isomorphic to the ring for any basic irreducible polynomial of degree over . It is clear that , that is, a finite field of elements, if , and , that is the ring of residue classes of modulo its ideal , if .

We denote by the group of units of . is a local ring with the maximal ideal , and all ideals of are given by . By [1, Theorem 14.8], there exists an element of multiplicative order , which is a root of a basic primitive polynomial of degree over and dividing in , and every element can be written uniquely as where . Moreover, is a unit if and only if , and is a zero divisor or 0 if and only if . Define the -exponent of by and if with . By [1, Corollary 14.9], , where is the cyclic group of order , and is the one group of Galois ring , so .

For a fixed positive integer , let and be the set of all matrices and the multiplicative group of all invertible matrices over , and denote by and the identity matrix and zero matrix, respectively. In this paper, for matrix and matrix over , we adopt the notation which is a matrix over .

For any matrix , is said to be *symmetric* if , where is the transposed matrix of . We denote the set of all symmetric matrices over by . Then is a group under the addition of matrices. For any matrices , if there exists matrix such that , we say that is *cogredient* to over . It is clear that if and only if . So cogredience of matrices over is an equivalent relation on . If , we call the *cogredient classes* of containing over . Let , be all distinct cogredient classes of . As in [2] we define relations on by
Then the system is an association scheme of class on the set and denoted by .

Let stand for an odd prime number in the following. When , we know that the class number of is given by and the association scheme has been investigated in [2]. When , two kinds of cogredient standard forms of symmetric matrices over are given in [3, 4]. If , and (mod 4), a complex formula to count the number of all distinct cogredient classes of is given in [3], which shows that, for example,

if is odd and is odd, then where the meanings of and formulas for other cases are referred to [3].

Then two problems arise. Is there a simple and explicit formula to count the number of all distinct cogredient classes of ? For arbitrary Galois ring , in order to determine precisely the class number of the association scheme , we have to count the number of all distinct cogredient classes of .

In Section 2 we give two kinds of cogredient standard forms for every symmetric matrix over arbitrary Galois ring of odd characteristic. In Section 3 we obtain an explicit formula to count the number of all distinct cogredient classes of , which is simpler than that of [3] for the special case .

Now, we list some properties for the Galois ring which will be needed in the following sections. For general theory of Galois rings, one can refer to [1].

Lemma 1.1 (see [1, Theorem 14.11]). * where is a cyclic group of order , and is a group of order .*

Proposition 1.2. * is a subgroup of with index . **
For any , , and . **
For any and , there exists such that .*

* Proof . *In the notation of Lemma 1.1. Let be a generator of the cyclic group . Then is of order . Since is odd and is even, is of order and . Since is odd and is a commutative group of order by Lemma 1.1, for every , there exists a unique such that , so . Moreover, by Lemma 1.1 each can be uniquely expressed as where and . (i) For every where and , if and only if there exist and such that , which is then equivalent to that and . So if and only if by Lemma 1.1. Then and so . Hence, by group theory. (ii) Since , for any , we have and by group theory. So by the proof of (i). (iii) Let and . Then . From this and by Lemma 1.1, there exists a unique element such that . Now, let . Then satisfying .

Proposition 1.3. *Let . Then for any , there exist such that .*

* Proof. *Let . Suppose that . Then there exists such that . So . Since is odd and in , there exists such that . From we deduce , which is a contradiction. Hence . Therefore, () is a mapping from to . Suppose that . Then for , there exists such that , which implies that , and we get a contradiction. So there exists such that , that is, by Proposition 1.2. Then there exists such that , so , where .

#### 2. Cogredient Standard Forms of Symmetric Matrices

In this section, we give two kinds of cogredient standard forms of symmetric matrices over corresponding to that of cogredient standard forms of symmetric matrices over finite fields (see [5], or [6], Theorems 1.22 and 1.25).

*Notation 1. *For any nonnegative integer and , define

Lemma 2.1. *For any and , is cogredient to .*

*Proof. *Let . Then there exists such that , that is, . Since is an odd prime number, we have and so . Let . Since is a commutative ring, we have . Hence, . Then by and , we get
so is cogredient to .

Let . Then by Proposition 1.3 there exist such that . Let . Then and so . By , a matrix computation shows that . Hence, is cogredient to as well.

Then is cogredient to .

Lemma 2.2. *Let and . *(i)* If , then is cogredient to . *(ii)* If , then is cogredient to .*

*Proof. *We select and denote that . From we deduce . Hence . Then by , we see that is cogredient to .(i) By there exists such that . Then is cogredient to . If , is cogredient to by Lemma 2.1. If , there exists such that , so is cogredient to as well. Therefore, is cogredient to in this case.(ii) Let . Then by Proposition 1.2 there exists such that . Hence is cogredient to . If , there exists such that , so is cogredient to . If , then , and hence there exists such that , so . Hence, is cogredient to as well. Therefore, is cogredient to .

Lemma 2.3. *Let and , where , and . Then, One has the following. *(i)* is necessarily cogredient to either or . Moreover, these two matrices are not cogredient over . *(ii)* If is odd, then D is necessarily cogredient to either or . Moreover, these two matrices are not cogredient. If is even, then D is necessarily cogredient to either or . Moreover, these two matrices are not cogredient.*

* Proof. * (i) We may assume that and , where . Then is cogredient to . If is even, by Lemma 2.1โโ is cogredient to and hence is cogredient to . Now, let be odd. If , is obviously cogredient to . If , by Lemma 2.1โโ is cogredient to , and hence is cogredient to .

Suppose that is cogredient to over . Then there exists such that . From this and by , we obtain that , which is a contradiction. So and are not cogredient over .

(ii) We have one of the following two cases.(ii-1)Let be an odd number. Then is even and we have one of the following two cases. (ii-1-1)Let . Then is cogredient to by Lemma 2.2(i). From this and by (i) we deduce that is cogredient to when is cogredient to , or is cogredient to when is cogredient to . (ii-1-2) Let . Then we have one of the following two cases.() Let be even. Then is cogredient to by Lemma 2.1, so is cogredient to . Since is cogredient to by Lemma 2.2(ii), by (i) we see that: is cogredient to when is cogredient to , or is cogredient to when is cogredient to . () Let be odd. Then for some nonnegative integer and so . By Lemma 2.1 we see that is cogredient to , and is cogredient to . Hence is cogredient to , which is then cogredient to . Since is cogredient to by Lemma 2.2(ii), is cogredient to . Moreover, is cogredient to , which is then cogredient to . Since is cogredient to by Lemma 2.2(ii), is cogredient to . Therefore, is necessarily cogredient to either or by (i).(ii-2) Let be an even number. Then is also even and we have one of the following two cases.(ii-2-1) Let . Then for some and so is cogredient to . By Lemma 2.2(i) is cogredient to when is cogredient to , or is cogredient to when is cogedient to .(ii-2-2) Let . Then for some . By , we see that is cogredient to . Now, we have one of the following two cases.() Let be even. Then is cogredient to by Lemma 2.1 and so is cogredient to . From this and by Lemma 2.2(ii), we see that is cogredient to . Let . Since is cogredient to and is cogredient to by Lemma 2.2(ii), is cogredient to . Now, let . Since is even, is cogredient to by Lemma 2.1, so is cogredient to . Hence, is cogredient to , which is then cogredient to . Since is cogredient to , we see that is cogredient to by Lemma 2.2(ii). Therefore, is necessarily cogredient to either or by (i).() Let be odd. Then there exists nonnegative integer such that and so . Since is cogredient to by Lemma 2.1, is cogredient to , that is then cogredient to by Lemma 2.2(ii). Now, is cogredient to by Lemma 2.1, which is then cogredient to . Hence is cogredient to by Lemma 2.2(ii). Therefore, is necessarily cogredient to either or by (i).

Theorem 2.4. *Let . Then every symmetric matrix over is necessarily cogredient to one of the following matrices:
**
where , or for all , , and satisfy .*

* Proof. *The statement holds obviously if (corresponding to the case ) or . Now, let and . Then, there exist such that and . Let . Then , and there exists such that where and is a symmetric matrix over satisfying or for all , . By induction there exists such that , where and . Then satisfies .

Now, there must exist , and such that . Then and is cogredient to , where is a matrix over for all . Since is cogredient to for every by Lemma 2.3(i), we deduce that is cogredient to .

Theorem 2.5. *Let . Then every symmetric matrix over is necessarily cogredient to one of the following matrices:
**
where is a matrix over R such that is equal to either or when is odd, and is equal to either or when is even, for all ; , , and satisfy .*

* Proof. *It follows from Theorem 2.4 and the proof of Lemma 2.3(ii).

For any symmetric matrix , we call the *cogredient standard form of kind* (I) of if is cogredient to , and call the *cogredient standard form of kind* (II) of if is cogredient to .

#### 3. The Number of Cogredient Classes of Symmetric Matrices

In order to count the number of all distinct cogredient classes of symmetric matrices over , we show that every symmetric matrix over has only one cogredient standard form of kind () first, then the number of all distinct cogredient classes of symmetric matrices over is equal to the number of all cogredient standard forms of kind () by Theorem 2.4.

Theorem 3.1. *The number of all distinct cogredient classes of symmetric matrices over is given by the following: *(i)* If , then ; *(ii)* If , then .*

*Proof. *Let , where or for all , , , and satisfy . In the notation of Theorem 2.4, by [7, Theorem ], it follows that if is cogredient to over . Hence, every symmetric matrix over has only one cogredient standard form of kind ().

For any , denote that and . Then , if and, if . From this and by Theorem 2.4 it follows that . Therefore, if and, if .

In the notations of Section 1, we see that the class number of the association scheme is determined by . Then by Theorem 3.1, we have the following corollary.

Corollary 3.2. *The class number of the association scheme is given by the following.*(i)* If , then ; *(ii)* If , then .*

*Example 3.3. *Let be an odd prime number and . Then by Theorem 3.1 the number of all cogredient classes of symmetric matrices over Galois ring is given by . In fact, for a fixed element , all cogredient standard forms of kind (I) of symmetric matrices over are given by the following:
The number of all cogredient classes of symmetric matrices over is given by . In fact, all cogredient standard forms of kind (I) of symmetric matrices over are given by the following: where is one of matrices in (3.1), and

*Example 3.4. *Let be an odd prime number and . Then by Theorem 3.1 the number of all cogredient classes of symmetric matrices over Galois ring is given by ; the number of all cogredient classes of symmetric matrices over is given by .

#### Acknowledgment

This reaserach is supported in part by the National Science Foundation of China (No. 10971160) and Natural Science Foundation of Shandong provence (Grant No. ZR2011AQ004).