Invertible and irreducible elements
Oliver Pretzel · 1992
Abstract There are two special classes of elements in a Euclidean domain D that are in a sense diametrical opposites. The first class consists of those elements that have inverses in D. These elements, called invertible, are too nice for division with remainder to be of any use. They also have a way of slipping in and out of expressions involving products, because if a is invertible and ab = I, then for any elements x and y, xy = (xa)(by). For this reason highest common factors and prime factorization are unique only up to multiplication by invertible elements.