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

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 $a,b \in R$ .