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In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative ...
Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. For the items in commutative algebra (the theory of commutative rings), see Glossary of commutative algebra.
Ring: a semiring whose additive monoid is an abelian group. Division ring: a nontrivial ring in which division by nonzero elements is defined. Commutative ring: a ring in which the multiplication operation is commutative. Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element).
for every element a of the ring (again, if n exists; otherwise zero). This definition applies in the more general class of rngs (see Ring (mathematics) § Multiplicative identity and the term "ring"); for (unital) rings the two definitions are equivalent due to their distributive law.
The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X to produce an associative multiplication that distributes over the standard addition.
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In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables.