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In mathematics, more specifically abstract algebra, a finite ring is a ring (not necessarily with a multiplicative identity) that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history. As with finite groups, the complexity of the classification depends upon the complexity of the prime factorization of ''m''. If ''m'' is the square of a prime, for instance, there are precisely eleven rings having order ''m''. On the other hand, there can be only two ''groups'' having order ''m''; both of which are abelian. The theory of finite rings is more complex than that of finite abelian groups, since any finite abelian group is the additive group of at least two nonisomorphic finite rings: the direct product of copies of , and the zero ring. On the other hand, the theory of finite rings is simpler than that of not necessarily abelian finite groups. For instance, the classification of finite simple groups was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, any finite simple ring is isomorphic to the ring of ''n''-by-''n'' matrices over a finite field of order ''q''. The number of rings with ''m'' elements, for ''m'' a natural number, is listed under in the On-Line Encyclopedia of Integer Sequences. ==Enumeration== In 1964 David Singmaster proposed the following problem in the American Mathematical Monthly: "(1) What is the order of the smallest non-trivial ring with identity which is not a field? Find two such rings with this minimal order. Are there more? (2) How many rings of order four are there?" One can find the solution by D.M. Bloom in a two-page proof that there are eleven rings of order 4, four of which have a multiplicative identity. Indeed, four-element rings introduce the complexity of the subject. There are three rings over the cyclic group C4 and eight rings over the Klein four-group. There is an interesting display of the discriminatory tools (nilpotents, zero-divisors, idempotents, and left- and right-identities) in Gregory Dresden's lecture notes (see reference). The occasion of ''non-commutativity'' in finite rings was described in in two theorems: If the order m of a finite ring with 1 has a cube-free factorization, then it is commutative. And if a non-commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper triangular 2 × 2 matrix ring over the Galois field of the prime. The study of rings of order the cube of a prime was further developed in and . Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case. Definitive work on the isomorphism classes came with proving that for ''p'' > 2, the number of classes is 3''p'' + 50. There are earlier references in the topic of finite rings, such as Robert Ballieu and Scorza.〔Scorza (1935), see review of Ballieu by Irving Kaplansky in Mathematical Reviews〕 These are a few of the facts that are known about the number of finite rings (not necessarily with unity) of a given order (suppose ''p'' and ''q'' represent distinct prime numbers): *There are two finite rings of order ''p''. *There are four finite rings of order ''pq''. *There are eleven finite rings of order ''p''2. *There are twenty-two finite rings of order ''p''2''q''. *There are fifty-two finite rings of order eight. *There are 3''p'' + 50 finite rings of order ''p''3, ''p'' > 2. The number of rings with ''n'' elements are (start with ''n'' = 0) :1, 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, 390, 2, 22, 2, 22, 4, 4, 2, 104, 11, 4, 59, 22, 2, 8, 2, >18590, 4, 4, 4, 121, 2, 4, 4, 104, 2, 8, 2, 22, 22, 4, 2, 780, 11, 22, ... 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「finite ring」の詳細全文を読む スポンサード リンク
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