Factorial ring

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.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)

Comments

For the notion of height see Height of an ideal.

A Zariski ring is a Noetherian ring $R$ having an ideal $\mathfrak a$ such that every ideal in $R$ is closed in the $\mathfrak a$-adic topology (cf. Adic topology). The last condition can be replaced by: Every element $r\in R$ for which $1-r\in\mathfrak a$ is invertible in $R$. A Zariski ring $(R, \mathfrak a)$ is complete if it is a complete topological space (in the $\mathfrak a$-adic topology). The completion of a Zariski ring $(R,\mathfrak a)$ is the completion of the topological space $R$ (in the $\mathfrak a$-adic topology). This completion, $\bar R$, is a Zariski ring (take $\mathfrak a\bar R$ as ideal).