ring

A ring is a set $R$ together with two binary operations, denoted $+:R\times R⟶R$ and $\cdot :R\times R⟶R$, such that

1. $(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for all $a,b,c\in R$ (associative law)

2. $a+b=b+a$ for all $a,b\in R$ (commutative law)

3. There exists an element $0\in R$ such that $a+0=a$ for all $a\in R$ (additive identity)

4. For all $a\in R$, there exists $b\in R$ such that $a+b=0$ (additive inverse)

5. $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ and $(a+b)\cdot c=(a\cdot c)+(b\cdot c)$ for all $a,b,c\in R$ (distributive law)

Equivalently, a ring is an abelian group $(R,+)$ together with a second binary operation $\cdot$ such that $\cdot$ 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 $1\in R$ such that $a\cdot 1=1\cdot a=a$ for all $a\in R$. 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 $a\in R$ is an element $b\in R$ such that $a\cdot b=b\cdot a=1$. An element of $R$ that has a multiplicative inverse is called a unit of $R$.

A ring $R$ is commutative if $a\cdot b=b\cdot a$ for all $a,b\in R$.