Ring (mathematics)
algebraic structure that has compatible structures of an abelian group and a monoid, in particular having multiplicative identity
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like addition and multiplication of integers. They work similarly to integer addition and multiplication, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
More formally, a ring is a set that is endowed with two operations, called addition and multiplication, such that the ring is an abelian group with respect to addition. The multiplication is associative, is distributive over the addition, and has a multiplicative identity. Some authors omit the requirement for a multiplicative identity in the definition of a ring, and call ring with identity the above defined structure; see below§ Variations in terminology for details.
A commutative ring is a ring with a commutative multiplication. This property has profound implications on ring properties. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry.
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This entry incorporates text from “Ring (mathematics)” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.