ring

A ring is an algebraic structure consisting of a set with two binary operations, typically denoted and used like the operations of addition and multiplication.

Formally,
A ring is a set $R$ equipped with two binary operations, addition and multiplication, such that for all elements $a,b,c\in R$,

$(R,+)$ is an abelian group:

1. $(R,+)$ is closed: $a+b\in R$
2. $(R,+)$ is associative: $(a+b)+c=a+(b+c)$
3. $(R,+)$ has an identity element: $a+0=a=0+a$
4. $(R,+)$ has inverses: $a+(-a)=0=(-a)+a$
5. $(R,+)$ is commutative: $a+b=b+a$

$(R,\times )$ is a monoid:

6. $(R,\times )$ is closed: $ab\in R$
7. $(R,\times )$ is associative: $(ab)c=a(bc)$
8. $(R,\times )$ has an identity element: $a1=a=1a$

Multiplication is distributive over addition:

9. $(R,+,\times )$ is left-distributive: $a(b+c)=ab+ac$
10. $(R,+,\times )$ is right-distributive: $(a+b)c=ac+bc$

A ring where $(R,\times )$ is an abelian group is a field.