Ring

A ring is a set $R$ equipped with two binary operations $+$ and $\cdot$ such that:

1. $(R,+)$ is an abelian group (in particular, there is an additive identity $0$ and every element has an additive inverse),
2. multiplication is associative: $(ab)c=a(bc)$ for all $a,b,c\in R$,
3. multiplication distributes over addition: $a(b+c)=ab+ac$ and $(a+b)c=ac+bc$ for all $a,b,c\in R$.

These conditions are often summarized as the ring axioms . Many texts additionally impose a multiplicative identity, leading to a unital ring ; commutativity of multiplication leads to a commutative ring .

Examples:

• $\mathbb Z$ with usual $+$ and $\cdot$ is a (unital, commutative) ring.
• $M_n(\mathbb Z)$ is a ring under matrix addition and multiplication, typically noncommutative for $n\ge 2$.
• The even integers $2\mathbb Z$ form a ring (under inherited operations) but are not unital.