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Finite ring
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 do, 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 – the n-by-n matrices over a finite field of order q (as a consequence of Wedderburn's theorems, described below).
The number of rings (meaning rngs) with m elements, for m a natural number, is listed under OEIS: A027623 in the On-Line Encyclopedia of Integer Sequences (OEIS). For the number of rings with multiplicative identity (refer to variations on terminology on the page ring), it is listed under OEIS: A037291. The number of rings/rngs are non-trivial problems in mathematics; both problems are listed as "hard" sequences in OEIS.
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:
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 dimension n over F. The matrix ring A of n-by-n matrices with elements from F is used in Galois geometry, with the projective linear group serving as the multiplicative group of A.
Wedderburn's little theorem asserts that any finite division ring is necessarily commutative:
Nathan Jacobson later discovered yet another condition that guarantees commutativity of a ring: if for every element r of R there exists an integer n > 1 such that r n = r, then R is commutative. More general conditions that imply commutativity of a ring are also known.
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Finite ring
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 do, 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 – the n-by-n matrices over a finite field of order q (as a consequence of Wedderburn's theorems, described below).
The number of rings (meaning rngs) with m elements, for m a natural number, is listed under OEIS: A027623 in the On-Line Encyclopedia of Integer Sequences (OEIS). For the number of rings with multiplicative identity (refer to variations on terminology on the page ring), it is listed under OEIS: A037291. The number of rings/rngs are non-trivial problems in mathematics; both problems are listed as "hard" sequences in OEIS.
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:
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 dimension n over F. The matrix ring A of n-by-n matrices with elements from F is used in Galois geometry, with the projective linear group serving as the multiplicative group of A.
Wedderburn's little theorem asserts that any finite division ring is necessarily commutative:
Nathan Jacobson later discovered yet another condition that guarantees commutativity of a ring: if for every element r of R there exists an integer n > 1 such that r n = r, then R is commutative. More general conditions that imply commutativity of a ring are also known.