Ring axioms

The ring axioms specify a set $R$ equipped with two binary operations $+$ and $\cdot$ such that:

1. $(R,+)$ is an abelian group (with identity element $0$).
2. Multiplication is associative: $(ab)c=a(bc)$ for all $a,b,c\in R$.
3. Distributive laws hold: $a(b+c)=ab+ac$ and $(a+b)c=ac+bc$ for all $a,b,c\in R$.

These axioms define a ring (not necessarily unital , and not necessarily commutative ). Most structural notions—such as an ideal and a ring homomorphism —are formulated relative to this axiomatic package.