Seja A {\displaystyle A} um conjunto com as operações internas ( + ) : ( a , b ) b → a + b {\displaystyle (+):(a,b)b\rightarrow a+b} e ( . ) : ( a , b ) → a . b {\displaystyle (.):(a,b)\rightarrow a.b} . ( A , + , . ) {\displaystyle (A,+,.)} é um anel se, ∀ a , b , c ∈ A : {\displaystyle \forall a,b,c\in A:}
1. ( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)} 2. ∃ 0 ; a + 0 = 0 + a = a {\displaystyle \exists 0;a+0=0+a=a} 3. ∀ x ∈ A , ∃ ! − x ∈ A ; x + ( − x ) = 0 {\displaystyle \forall x\in A,\exists !-x\in A;x+(-x)=0} 4. a + b = b + a {\displaystyle a+b=b+a} 5. ( a . b ) . c = ( a . b ) . c {\displaystyle (a.b).c=(a.b).c} 6. a . ( b + c ) = a . b + b . c e ( b + c ) . a = b . a + c . a {\displaystyle a.(b+c)=a.b+b.ce(b+c).a=b.a+c.a}