Commutative ring

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{math, number, function}

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra.

Some specific kinds of commutative rings are given with the following chain of class inclusions:

Contents

Definition and first examples

Definition

A ring is a set R equipped with two binary operations, i.e. operations that combine any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by "+" and "⋅", e.g. a + b and ab. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, such that multiplication distributes over addition, i.e. a ⋅ (b + c) = (ab) + (ac). The identity elements for addition and multiplication are denoted 0 and 1, respectively.

If, in addition, the multiplication is also commutative, i.e.

the ring R is called commutative. In the remainder of this article, all rings will be commutative, unless explicitly stated otherwise.

First examples

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