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.
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L'étudiant acquiert une initiation théorique à la méthode des éléments finis qui constitue la technique la plus courante pour la résolution de problèmes elliptiques en mécanique. Il apprend à applique
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In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is (F ∖ {0}, •), where 0 refers to the zero element of F and the binary operation • is the field multiplication, the algebraic torus GL(1).. The multiplicative group of integers modulo n is the group under multiplication of the invertible elements of .
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power.
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. It is then called a simple algebra over this field. Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple).
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