Irreducible but not prime：Irreducible and not prime [duplicate]

Here is an example that is perhaps simpler than the common quadratic integer examples. $\:$ Let $\rm R = \mathbb R + x\:\mathbb C[x],\:$ i.e. the ring of all polynomials with complex coefficients and real constant coefficient. Here $\rm\:x2\:$ has infinitely many distinct factorizations into irreducibles

$$\rm x2\ =\ (c\: x)\: (c{-1}\: x),\quad c = r + {\it i},\quad \forall\: r\in \mathbb R$$

The factors are nonassociate irreducibles in $\rm R$ since, for $\rm\:r,s\in \mathbb R$

$$\rm (r+{\it i})x\ |\ (s+{\it i})x\ \ in\ \ R\iff \frac{(s+{\it i})\:x}{(r+{\it i})\:x}\in R\iff \frac{s+{\it i}}{r+{\it i}}\in \mathbb R\iff r = s$$

In particular, the irreducible $\rm\,x\,$ is not prime.

Note $\:$ Such constructions are often used by ring theorists since they yield a very rich source of (counter-) examples, e.g. see $\ $ M. Zafrullah, Various facets of rings between $\rm\:D[X]\:$ and $\rm\:K[X].$[1]

References Various facets of rings ...