# Atomic ring

An integral domain with a unit element, satisfying the ascending chain condition (maximum condition) for principal ideals. In other words, any family of principal ideals of an atomic ring has a maximal element. An element of the ring is called an atom, or an extremal element, or an indecomposable element, if it cannot be decomposed into a product of non-invertible elements. An integral domain is an atomic ring if and only if each non-zero non-invertible element is a product of atoms. All Noetherian rings are atomic. Factorial rings are atomic rings in which each atom is a prime element (i.e. an element that generates a prime ideal). An atomic Bezout ring is a principal ideal ring.

#### Comments

There seems to be no special name established in Western literature for these rings. Instead one simply speaks of rings which satisfy the ascending chain condition for principal ideals or the divisor chain condition.

#### References

[a1] | R. Gilmore, "Multiplicative ideal theory" , M. Dekker (1972) |

**How to Cite This Entry:**

Atomic ring. V.I. Danilov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Atomic_ring&oldid=15746