Introduction to Groups, Rings and Fields

Addition: for each pair of real numbers a and b there exists a unique real number a + b such that • + is a commutative and associative operation; • there exists in R a zero, 0, for addition: a+ 0 = 0 + a = a for all a ∈ R; • for each a ∈ R there exists an additive inverse −a ∈ R such that a+(−a) = (−a)+a = 0. Multiplication: for each pair of real numbers a and b there exists a unique real number a · b such that • · is a commutative and associative operation; • there exists in R an identity, 1, for multiplication: a · 1 = 1 · a = a for all a ∈ R; • for each a ∈ R with a 6= 0 there exists an additive inverse a ∈ R such that a · a = a · a = 1. Addition and multiplication together: for all a, b, c ∈ R, we have the distributive law a · (b+ c) = a · b+ a · c. Avoiding collapse: we assume 0 6= 1.