Unique factorization domains

Throughout this chapter R is a commutative integral domain with unity. Such a ring is also called a domain.

If a and b are nonzero elements in R, we say that b divides a (or b is a divisor of a) and that a is divisible by b (or a is a multiple of b) if there exists in R an element c such that a = bc, and we write b|a or a ≡ 0 (mod b). Clearly, an element u ∈ R is a unit if and only if u is a divisor of 1.

Two elements a,b in R are called associates if there exists a unit u ∈ R such that a = bu. Of course, then b = av, where v = u-1. This means that if a and b are associates, then a|b and b|a. In fact, if R is a commutative integral domain, then the converse is also true; that is, if a|b and b|a, then a and b are associates. For let b = ax and a = by. Then b = byx. Because R is an integral domain with 1, this gives yx = 1. Therefore x and y are units, and, hence, a and b are associates.

We call an element b in R an improper divisor of an element a ∈ R if b is either a unit or an associate of a.