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Journal of Mathematics
Volume 2013 (2013), Article ID 467905, 8 pages
Classification of Finite Rings of Order by Generators and Relations
1Department of Mathematics, Islamic Azad University, Khorramabad Branch, Khorramabad 68178-16645, Iran
2Department of Mathematics, Lorestan University, P.O. Box 465, Khorramabad 68137-17133, Iran
3Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord 88186-34141, Iran
Received 26 September 2012; Accepted 10 December 2012
Academic Editor: Liwei Zhang
Copyright © 2013 P. Karimi Beiranvand et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
For any finite abelian group , we define a binary operation or “multiplication” on and give necessary and sufficient conditions on this multiplication for to extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there are rings of order with characteristic , where is a prime number. Also, all finite rings of order are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.
The problem of determining and classifying up to isomorphism finite rings has received considerable attention, both old and new, see [1–4]. It is well known that if is a finite ring (with identity), then the additive group of splits as the direct of its -primary components , where is a prime number, and these are pairwise orthogonal ideals. Thus is the direct sum of the rings . We provide a new proof of this fact in the paper.
Now let be a prime number. It is easy to see that there are two rings of order , and the null ring of order . In , Raghavendran (1969) proved that there exist eleven rings of order , only four of these rings have identity. Also in , Gilmer and Mott (1973) showed that there exist rings of order , only twelve of these rings have identity, and 59 rings of order 8, only eleven of these rings have identity. For , a comprehensive list of noncommutative rings was first only drawn up in . Commutative rings of order have been characterized by Wilson . Finally, Corbas and Williams (2000) in [1, 8], determined all rings of order . The rings of upper order are not still characterized.
Throughout the paper, all rings are associative (not necessarily commutative or with identity). For a set , denotes the cardinal of . For two integers and positive integer , we denote in case and are congruent modulo , and in case divides . Also, for two integers and , gcd denotes the greatest common divisor of and . In , the authors introduced a multiplication (similar to Theorem 3) on a finitely generated -module to extend to an -algebra, where is a commutative ring. Also a number of necessary and sufficient conditions for any -module to extend to an -algebra, where is a commutative ring, is given by Behboodi et al. . Here instead of modules over a commutative ring, we focus our attention to the -modules. In Section 2, by using the method in , we define a binary operation or “multiplication" on a finite abelian group and extend to a ring. In particular, we prove that there are rings (up to isomorphism) of order whose abelian group is cyclic. Also, all rings of order whose abelian group is are determined. In Section 3, an algorithm for the computation of all finite rings based on their additive group is given.
2. A Representation of Finite Rings and Fundamental Theorems
We begin the paper with the following well known fact and give a new proof of this fact.
Theorem 1. Let be a ring of order , where the are distinct primes and the are positive integers. Then is expressible, in a unique manner, as the direct sum of rings , where .
Proof. We know that the additive group is expressible, in a unique manner, as the direct sum of groups , where . We claim that is an ideal of , for . Suppose that and where , for . Then + + + + + . Since , . If , then we must have , a contradiction. Thus and so . Now suppose that and where and . We show that . Let , where , for . Then implies that and hence for . Now for any , if , then , a contradiction. Thus for all , . On the other hand, since , by a similar argument, we conclude that for all . Thus and so for all , is an ideal of , and the proof is complete.
Remark 2. If is a finite ring, then its additive group is a finite abelian group and is thus a direct product of cyclic groups. Suppose these have generators of orders . Then the ring structure is determined by the products
and thus by the structure constants . As  we introduce a convenient notation, motivated by group theory, for giving the structure of a finite ring. A presentation for a finite ring consists of a set of generators of the additive group of together with relations. The relations are of two types: (i) for ; (ii) with for .
If the ring has the presentation above we write
Theorem 3. Let be a finite ring with additive group , where are cyclic subgroups of orders , respectively. Assume , , where , and “” is the following operation Then (1)“” is well-defined if and only if (2)“” is always distributive, in the case when “” is well-defined;(3)“” is associative if and only if Consequently, is a ring if and only if and above hold.
Proof. (1) (). Assume “” is well-defined and fix where . Let . Then by definition of “”, we have
Thus for each , we deduce , for . Similarly, , for . Therefore,
Now one can easily check that the last relation holds if and only if holds.
(1) (). Assume where . Then and for all . Let and . Then , , and . Hence and so Now by condition we have It follows that that is, Thus the operation “” is well-defined.
(2) Suppose the operation “” is well-defined. Then
On the other hand Thus “” is a distributive operation.
(3) By definition of “”, we have and so and also Now, it is clear that “” is associative if and only if for all , if and only if
Theorem 4. For , let be two presentations with suitable , . Assume where and is the following map: Then (1) is well-defined if and only if for , (2)If is well-defined, then(a) is an -module homomorphism,(b) is one to one if and only if for each implies that for all , and (c) is a ring homomorphism if and only if for ,
Proof. (1) (). Assume , where . Then by definition we have
It follows that for all . Therefore, , for all and so for ,
(1) (). Suppose where . Then for all ) and so . Now by our hypothesis for all . Therefore, This means that and hence is well-defined.
(2) (a) Assume , for . Clearly, Thus is an -module homomorphism.
(2) (b) By definition , (b) is clear.
(2) (c) By (2) (a), is always a group homomorphism. Thus it is sufficient to show that for each . However by multiplications of , and definition of we have and also Thus is a ring homomorphism if and only if for all ,
Corollary 5. Let be a positive integer and let be cyclic additive groups of orders , respectively. If are two presentations with suitable , then if and only if there exist () such that(1), for ; (2)for all and , implies that for all , and (3), for .
Proof. It is clear by Theorem 4.
3. Finite Rings of Order with Characteristic and
In this section, we first classify all finite rings of order with characteristic . Then this classification is extended to the rings order with characteristic .
(i) Let be an additive cyclic group of order and . Then is always a ring.
(ii) Let and be two presentations with . Assume and is the following map: Then (1) is well-defined;(2) is a ring homomorphism if and only if for ;(3) is one to one if and only if for each implies that .
Since the function introduced in preceding proposition is multiplication by some , and is always well-defined, we have the following corollary.
Corollary 7. Let and be two presentations with . Then if and only if there exists such that(1);(2)for each implies that .
Proof. It is clear by Proposition 6.
Theorem 8. For any prime number , there are exactly seven rings of order whose additive group is cyclic, only one of these rings have identity. However all of these are commutative.
Proof. Let and be two presentations with . The relation in Corollary 7 implies that is relatively prime to and so by the relation in Corollary 7, we deduce that . Since the relation has a solution, gcd. Now we proceed by cases.
Case 1. gcd, that is, is relatively prime to . Since gcd, is also relatively prime to . Thus where both and are relatively prime to .
Case 2. gcd. Then the relation shows that either gcd or . But the relation follows that and hence because gcd. Since gcd, . Therefore we obtain that and so gcd. Thus where both gcd and gcd.
Case 3. gcd. Then the relation gcd shows that either gcd or . But the relation follows that and hence because gcd. Since gcd, . Therefore we obtain that and so gcd. Thus where both gcd and gcd.
Case 4. gcd. Then the relation gcd shows that either gcd or But the relation follows that and hence because gcd. Since gcd, . Therefore we obtain that and so gcd. Thus where both gcd and gcd.
Case 5. gcd. By a similar argument, we deduce where both gcd and gcd.
Case 6. gcd. By a similar argument, we deduce where both gcd and gcd.
Case 7. gcd. By a similar argument, we deduce where both gcd and gcd.
An analysis similar to that in the proof of Theorem 8 shows that there are exactly rings of order with characteristic . These rings are
We can obtain the following result by applying relations and in Theorem 3, for the rings of order with characteristic . Also, this manner may be used for obtaining all finite rings of order with characteristic , where .
Corollary 9. Let where A1 = <a1> and A2 = <a2> be cyclic groups of order and , respectively. Assume where . Then is a ring if and only if the following hold.(1) and ; (2);(3);(4);(5);(6).
Application and simplification of the relations (1), (2), and (3) in Corollary 5, for the rings of order with characteristic , give us the following result.
Corollary 10. Let be two rings with suitable integers and satisfied in Theorem 3, for . Then if and only if there exist such that the following hold.(1); (2)If and , then and ; (3); (4); (5); (6); (7); (8); (9).
In the following, one gives an algorithm for computing the finite rings whose abelian group is isomorphic to . We represent the algorithm in the form of two sub-algorithms. In Algorithm 11, one lets be the input and be the set of all sequences that satisfy the condition , for any . Then one has the sequences that satisfy the conditions and in Theorem 3 as the outputs of the algorithm. In fact, any sequence that is an output of this algorithm shows a ring as the following: In Algorithm 12, one represents the conditions required for the isomorphism of two rings based upon Theorem 4. In fact, one verifies the two sequences and obtained from Algorithm 11, under isomorphic conditions in Theorem 4. Therefore under Algorithms 11 and 12, one computes all the rings whose abelian group is isomorphic to and are not isomorphic. We have implemented Algorithms 11 and 12 in and presents a sample of its responses for the rings whose abelian group is isomorphic to in the following example.
Algorithm 11. (1) Input
(3) For all , let
(4) Let ; for all
(6) If , then go to end
(9) If there is such that , then go to step 6
(10) If there is such that , then go to step 6
Algorithm 12. (1) Input
(3) For all , let
(4) Input and satisfied in Algorithm 11
(5) Let ; for all
(7) If , then go to end
(10) If there is such that , then go to step 7
(12) Let ; for all
(15) If there is such that , then go to step 7
(16) If there is such that , then go to step 7
(17) Print two rings and with representations and , respectively, are isomorphic
Example 13. We compute the presentations of all finite rings of order , whose abelian group is isomorphic to . Let be a such ring. Then we write the following presentation for :
The outputs of these algorithms for the rings of order whose abelian group is are listed below.
Thus there exist rings of order with characteristic .
This research was supported by Islamic Azad University, Khorramabad Branch.
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