Is there a ring $R \neq \mathbb{Z}$ such that the fundamental theorem of arithmetic holds in $R$? Is there a ring, unequal to $\mathbb{Z}$, for which the fundamental theorem of arithmetic holds?
Please also give an example if an existential proof is given.
 A: There are many simple examples of Unique Factorization Domains (UFDs), e.g. the subring of $\,\Bbb Q\,$ of rationals with odd least denominator has only one prime $\,2,\,$ and every nonunit $\ne 0$ has a unique prime factorization $\, u\, 2^n\,$ for some unit (invertible) $\,u.\,$ Parity arguments extend to this ring.
More generally, Euclidean domains are those domains enjoying division with smaller remainder, and essentially the same proof as in $\,\Bbb Z\,$ shows that they are UFDs. Well-known simple examples are domains $\,F[x]\,$ of polynomials over a field, where the division algorithm is high-school polynomial long division. Another simple example is the domain of Gaussian integers $\,\Bbb Z[i].\,$
A: In a factorization domain, also known as an atomic domain, all elements have factorizations into products of irreducible elements, although not necessarily uniquely. Note how the word "atom" fits in this setting: they are the building blocks everything is composed of. Indeed, if we consider equivalence classes of elements (under the relation of being associate by a unit) to be partially ordered by the divisibility relation then the irreducible elements are the atoms.
Any number ring ${\cal O}_K$ is a factorization domain (number rings are special cases of Dedekind domains which are Noetherian which implies factorization). However they are not necessarily unique factorization domains (UFDs) - factorizations into irreducibles may not be unique (or even have the same length). If one changes from the element POV to the ideal POV, one recovers unique factorization for Dedekind domains as a property of ideals rather than numbers (which is a defining feature).
There is a difference between an irreducible element and a prime element. In a UFD they coincide, but not generally. Any factorization into primes is automatically unique up to ordering and associates, but factorizations into prime elements needn't exist. Primes are always irreducible but not generally vice-versa. The ideal analogue is the distinction between prime ideals and maximal ideals. Maximal ideals are prime but not generally vice-versa, and they coincide in Dedekind domains.
