Ring

A Set $(R,+,\times )$ with Addition and Multiplication operators such that:

1. Closed Under Addition ($\forall a,b\in R,a+b\in R$)
2. Closed Under Multiplication $\forall a,b\in R,a\times b\in R$

Axioms

The axioms are:

1. Additive Identity ($\exists 0\in R\text{ s.t }\forall a\in R,0{+}_{R}a=a$)
2. Additive Inverse exists ($\forall a\in R,\exists -a\in R\text{ s.t }a+(-a)=0$)
3. Associative with Addition ($\forall a,b,c\in R,(a{+}_{R}b){+}_{R}c=a{+}_R(b{+}_{R}c)$)
4. Commutative under Addition $(\forall a,b\in R,(a{+}_{R}b)=(b{+}_{R}a))$
5. Multiplicative Identity exists ($\exists 1\in R,\forall r\in R,1\ast r=r$)
6. Associative with Multiplication ($\forall a,b,c\in R,(ab)c=a(bc)$)
7. Distributive under Addition ($\forall r,s,w\in R,(r+s)(w)=rw+rs$)