Irreducible not implies universal side divisor

Statement
An fact about::irreducible element in an integral domain need not be a fact about::universal side divisor. In fact, an irreducible element need not be a universal side divisor even in a fact about::Euclidean domain.

Converse

 * Universal side divisor implies irreducible: In an integral domain, any universal side divisor is irreducible.
 * Universal side divisor not implies prime: In an integral domain, a universal side divisor need not be prime.

Caveats
There are some Euclidean domains where every irreducible element is a universal side divisor. For instance:


 * Suppose $$k$$ is an algebraically closed field. Then, the polynomial ring $$k[x]$$ has the property that every irreducible polynomial is a universal side divisor. This is because in a polynomial ring, the universal side divisors are precisely the nonconstant linear polynomials, and the field being algebraically closed is precisely equivalent to saying that these are the only irreducible polynomials.
 * In a discrete valuation ring, the uniformizing parameter, which is the unique irreducible (up to associates) is a universal side divisor. (Note that since both the property of being irreducible and the property of being a universal side divisor are preserved up to associates in an integral domain, this makes sense).

Example of the ring of integers
In the ring of rational integers $$\mathbb{Z}$$, the only universal side divisors are the elements $$\pm 2, \pm 3$$. However, there are infinitely many primes. In particular, $$\pm 5$$ are irreducibles that are not universal side divisors.

Example of the polynomial ring over a field that is not algebraically closed
Let $$k$$ be a field that is not algebraically closed. In the polynomial ring $$k[x]$$, the universal side divisors are precisely the nonconstant linear polynomials. However, since $$k$$ is not algebraically closed, there exists an irreducible polynomial of degree greater than $$1$$ in the polynomial ring $$k[x]$$. Thus, there is an irreducible element that is not a universal side divisor.

For instance, if $$k = \R$$, the polynomial $$x^2 + 1 \in \R[x]$$ is irreducible but not a universal side divisor.