Page 80 -

§1. Divisibility Properties of Factorial Rings 57

Appendix to Chapter 2
Factorial Rings and Elimination Theory

Factorial rings, also called unique factorization domains, have most of the strong
divisibility properties of the ring of integers Z. These properties are used in Chapter
5 for the study of the reduction of a plane curve modulo a prime number. The concept
is also useful for the understanding of divisibility properties of polynomials over a
field. Included in our brief introduction to the theory of these rings is a discussion of
the resultant of two polynomials which was used in Chapter 2, §§2 and 3.
All rings considered in this appendix are commutative.

§1. Divisibility Properties of Factorial Rings
A unit in a ring R is an element u ∈ R such that there exists an element v ∈ R with
∗
uv = 1. The units in R form a group under multiplication which we denote by R .
∗
Note that R is a field if and only if R = R −{0}.
(1.1) Definition. For a, b in R the element a divides b, denoted a|b, provided there
exists x in R with b = ax. In terms of ideals this condition can be written Ra ⊃ Rb.

Observe that u|a for any unit u and a|0 for all a in R. Moreover, Ra = Rb if and
only if b = ua, where u is a unit.

(1.2) Definition. A nonzero element p in R is an irreducible provided for each fac-
torization p = ab either a or b is a unit but not both.

Since each factorization of a unit is by units, an irreducible is not a unit. In a field
there are no irreducibles.
(1.3) Definition. A factorial ring R is an integral domain such that for any nonzero
a in R we can decompose a as
a = up 1 ,... , p r ,

where u is a unit and p 1 ... , p r are irreducibles, and also this factorization is unique
in the following sense: for a second decomposition a = vq i ,... , q s by irreducibles
we have r = s and after permutation of the q i ’s each p i = u i q i , where u i is a unit in
R.

(1.4) Alternative Formulations. Let R be a ring. The following remarks give con-
ditions under which unique factorization is possible.

(a) Every nonzero element of a ring R can be factored as a product of irreducible
elements if and only if every sequence of principal ideals

Ra 1 ⊂ Ra 2 ⊂· · ·
is stationary, that is, there exists m with Ra m = Ra m+1 =· · · .