Atomic domain

Remarks

Example: $\Z[\sqrt{-5}]$ is atomic, because it is Noetherian. In a Noetherian domain, a nonzero nonunit without a factorization would give an infinite strictly ascending chain of principal ideals. Non-example: the ring $\overline{\Z}$ of all algebraic integers is not atomic. For a nonzero nonunit $\alpha$, a square root $\beta$ of $\alpha$ is an algebraic integer and a nonunit, and $\alpha = \beta^2$. So no element of $\overline{\Z}$ is irreducible, and $2$ has no factorization. A domain that satisfies the ascending chain condition on principal ideals is atomic; the converse is false. The factorization is not required to be unique; uniqueness up to order and units is what makes $R$ a unique factorisation domain.