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Concept
Finite ring
Summary
In mathematics, more specifically abstract algebra, a finite ring is a ring 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.
Although rings have more structure than groups, the theory of finite rings is simpler than that of 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, it has been known since 1907 that any finite simple ring is isomorphic to the ring of n-by-n matrices over a finite field of order q (as a consequence of Wedderburn's theorems, described below).
The number of rings with m elements, for m a natural number, is listed under in the On-Line Encyclopedia of Integer Sequences.
Finite field and Finite field arithmetic
The theory of finite fields is perhaps the most important aspect of finite ring theory due to its intimate connections with algebraic geometry, Galois theory and number theory. An important, but fairly old aspect of the theory is the classification of finite fields:
The order or number of elements of a finite field equals pn, where p is a prime number called the characteristic of the field, and n is a positive integer.
For every prime number p and positive integer n, there exists a finite field with pn elements.
Any two finite fields with the same order are isomorphic.
Despite the classification, finite fields are still an active area of research, including recent results on the Kakeya conjecture and open problems regarding the size of smallest primitive roots (in number theory).
A finite field F may be used to build a vector space of n-dimensions over F. The matrix ring A of n × n matrices with elements from F is used in Galois geometry, with the projective linear group serving as the multiplicative group of A.
Official source
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