Commutative ring

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. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not or is not required to be commutative.

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

A ring is a set R equipped with two binary operations, i.e. operations combining 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 a ⋅ b. 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, where multiplication distributes over addition; i.e., a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c). The identity elements for addition and multiplication are denoted 0 and 1, respectively.

If the multiplication is commutative, i.e.

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

An important example, and in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication. As the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z as an abbreviation of the German word Zahlen (numbers).

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