A number system that is not unique factorization domain

Question

Can anyone present a number system that is not unique factorization domain and is a commutative ring?

So I want the case that does not involve polynomials/monomials or some trivial cases.

Answer 1

The ring $\mathbb{Z}[\sqrt{-5}]$ of complex numbers of the form $a+b\sqrt{-5}$ with $a,b\in \mathbb{Z}$ is not a UFD because $6=2\cdot 3$ and $6=(1+\sqrt{-5})(1-\sqrt{5})$; none of $2,3,1+\sqrt{-5},1-\sqrt{-5}$ are associates, so even when we make these irreducible factorizations (they actually are already), they won't be the same.

Answer 2

Consider the set of all multiples of 2. This is a commutative ring. 42, 66, 70, and 110 are all irreducible, and not associates of each other, and $4620=42\times110=66\times70$.

Answer 3

Hint $ $ In the subring of $\,\Bbb Q[x]\,$ of integer-valued polynomials, $\ 2\mid x(x\!+\!1),\,$ but $\, 2\nmid x, x\!+\!1$.