Factorial ring

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

where is invertible and the non-negative integral exponents are non-zero for only a finite number of elements . Here an element is called irreducible in if implies that either or is invertible in , and is not invertible in .

In a factorial ring there is a highest common divisor and a least common multiple of any two elements. A ring is factorial if and only if it is a Krull ring and satisfies one of the following equivalent conditions: 1) every divisorial ideal of 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 is a multiplicative system in a factorial ring , then the ring of fractions is factorial. A Zariski ring is factorial if its completion 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 having an ideal such that every ideal in is closed in the -adic topology (cf. Adic topology). The last condition can be replaced by: Every element for which is invertible in . A Zariski ring is complete if it is a complete topological space (in the -adic topology). The completion of a Zariski ring is the completion of the topological space (in the -adic topology). This completion, , is a Zariski ring (take as ideal).