rings | Guzman's Mathematics Weblog

Proofs from THE BOOK by Martin Aigner and Günter Ziegler begins by giving six proofs of the infinity of primes. We will go over the second proof. Before we go over this proof, lets cover some background.

— 1. Algebraic Structures —

One of the themes of modern algebra is to compare algebraic structures. An algebraic structure refers to a nonempty set equipped with a binary operation or several binary operations (usually two). Also, we shall refer to a nonempty set equipped with two binary operations satisfying certain properties as a number system. First, we shall define algebraic structures called group, ring and field. A group consists of nonempty set with one binary operation satisfying several conditions. It is the simplest of the three.

Definition 1 Let ${G}$ be a nonempty set and ${*:G\times G\rightarrow G}$ be a binary operation on ${G.}$ Then the set ${G}$ together with the binary operation ${*}$ , denoted ${(G,*)}$ , is a group, if the following three conditions are satisfied:

i. The binary operation ${*}$ is associative, i.e., ${(a*b)*c=a*(b*c)}$ for all ${a,b,c\in G}$ .