A ring is a set R together with two binary operations, denoted +:R×R⟶R and ⋅:R×R⟶R, such that
- (a+b)+c=a+(b+c) and (a⋅b)⋅c=a⋅(b⋅c) for all a,b,c∈R (associative law)
- There exists an element 0∈R such that a+0=a for all a∈R (additive identity)
- a⋅(b+c)=(a⋅b)+(a⋅c) and (a+b)⋅c=(a⋅c)+(b⋅c) for all a,b,c∈R (distributive law)
Equivalently, a ring is an abelian group
(R,+) together with a second binary operation ⋅ such that ⋅ is associative and distributes over +. Additive inverses are unique, and one can define subtraction in any ring using the formula a-b:=a+(-b) where -b is the additive inverse of b.
We say R has a multiplicative identity if there exists an element 1∈R such that a⋅1=1⋅a=a for all a∈R. Alternatively, one may say that R is a ring with unity, a unital ring, or a unitary ring. Oftentimes an author will adopt the convention that all rings have a multiplicative identity. If R does have a multiplicative identity, then a multiplicative inverse of an element a∈R is an element b∈R such that a⋅b=b⋅a=1. An element of R that has a multiplicative inverse is called a unit of R.
A ring R is commutative if a⋅b=b⋅a for all a,b∈R.