Factorization in Integral Domains I

Definition 1.1. For r, s ∈ R, we say that r divides s (written r|s) if there exists a t ∈ R such that s = tr. An element u ∈ R is a unit if it has a multiplicative inverse, i.e. if there exists an element v ∈ R such that uv = 1. The (multiplicative) group of units is denoted R∗. If r, s ∈ R, then r and s are associates if there exists a unit u ∈ R∗ such that r = us. In this case, s = u−1r, and indeed the relation that r and s are associates is an equivalence relation (HW). Another easy result is (this was on the midterm):