Irreducible but not prime element?

Question

>I am looking for a ring element which is irreducible but not prime. So necessarily the ring can't be a PID. My idea was to consider $R=K[x,y]$ and $x+y\\in R$. This is irreducible because in any product $x+y=fg$ only one factor, say f, can have a $x$ in it (otherwise we get $x^2$ in the product). And actually then there can be no $y$ in $g$ either because $x+y$ has no mixed terms. Thus $g$ is just an element from $K$, i.e. a unit. I got stuck at proving that $x+y$ is **not** prime. First off, is this even true? If so, how can I see it?

Answer 1

Hi! This isn't a "prealgebra" question, but here's an answer.

The example you chose, R = K[x,y], is an unfortunate one, because the polynomial ring is a UFD, so all irreducible elements are in fact prime--including (x+y). However, there are certainly rings where this is not true: in integral domains, all primes are irreducible but not all irreducible elements are prime. The wikipedia page on irreducible elements walks through an elementary example in the quadratic integer ring.