Commutative unital ring

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

Definition

A commutative unital ring is a set $R$ endowed with two binary operations $+$ and $*$, and constants $0$ and $1$ such that:

• $R$ is an Abelian group under $+$, with identity element $0$
• $R$ is an Abelian monoid under $*$, with identity element $1$
• Left and right distributivity laws hold:

$a * (b + c) = (a * b) + (a * c)$

and:

$(a + b) * c = (a * c) + (b * c)$