# Factorial ring

(Redirected from Unique factorization domain)

2010 Mathematics Subject Classification: Primary: 13F15 [MSN][ZBL]

unique factorisation domain, Gaussian ring

A ring with unique decomposition into factors. More precisely, a factorial ring $A$ is an integral domain in which one can find a system of irreducible elements $P$ such that every non-zero element $a\in A$ admits a unique representation

$$a=u\prod_{p\in P}p^{n(p)},$$

where $u$ is invertible and the non-negative integral exponents $n(p)$ are non-zero for only a finite number of elements $p\in P$. Here an element is called irreducible in $A$ if $p=uv$ implies that either $u$ or $v$ is invertible in $A$, and $p$ is not invertible in $A$.

In a factorial ring there is a highest common divisor and a least common multiple of any two elements. A ring $A$ is factorial if and only if it is a Krull ring and satisfies one of the following equivalent conditions: 1) every divisorial ideal of $A$ is principal; 2) every prime ideal of height 1 is principal; and 3) every non-empty family of principal ideals has a maximal element, and the intersection of any two principal ideals is principal. Every principal ideal ring is factorial. A Dedekind ring is factorial only if it is a principal ideal ring. If $S$ is a multiplicative system in a factorial ring $A$, then the ring of fractions $S^{-1}A$ is factorial. A Zariski ring $R$ is factorial if its completion $\hat R$ is.

Subrings and quotient rings of a factorial ring need not be factorial. The ring of polynomials over a factorial ring and the ring of formal power series over a field or over a discretely-normed ring are factorial. But the ring of formal power series over a factorial ring need not be factorial.

An integral domain is factorial if and only if its multiplicative semi-group is Gaussian (see Gauss semi-group), and for this reason factorial rings are also called Gaussian rings or Gauss rings.

How to Cite This Entry:
Unique factorization domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unique_factorization_domain&oldid=37552