276 Ring
A set with addition and multiplication linked by the distributive laws that is used to define polynomials.
definition (Ring) A set to which the following axioms apply:
- (Associativity) \(a+(b+c) = (a+b)+c\)
- (Commutativity) \(a+b = b+c\)
- (Zero) \(a+0 = 0\)
- (Additive Inverse) \(a+(-a) = 0\)
- (Associativity, Multiplication) \((ab)c = a(bc)\)
- (Commutativity, Multiplication) \(ab = ba\)
- (Unity) \(a1 = a\)
- (Distributivity) \(a(b+c)=a(b) + a(c)\)
where
- \(a, b, c\) are elements of the set.
- \(0\) is the additive identity.
- \(1\) is the multiplicative identity.
- \(-a\) is the additive inverse of \(a\).