# Ring where irreducibles are primes which is not an UFD

I am looking for

an example of a ring where unique factorization does not hold. However the ring is supposed to have the property that every irreducible is already a prime.

So unique factorization should fail because one cannot decompose every element into a finite product of irreducibles. In particular the ring must not be Noetherian. What is an example of such a ring?

-

How about $\mathbb C[X_1, X_2, \ldots]/(X_k-X_{k+1}^2\mid k\in\mathbb N)$? The only ireducible elements are the constants: Every polynomial in several variables $X_1,\ldots,X_n$ can be rewritten as one in $X_n$ only. Unless it is constant, we can even write it as a polynomial of degree $\ge 2$ in $X_{n+1}$ and this has a nontrivial linear factor $X_{n+1}-\alpha$. Since the product of constants is constant, not every element can be written as product of irreducibles.
In the ring of all algebraic integers, there are no irreducibles (since, for example, if $a$ is an algebraic integer, so is $\sqrt a$, and $a=\sqrt a\sqrt a$), so every irreducible is (vacuously) prime. You cannot decompose any element into a finite product of irreducibles since, as noted, there aren't any irreducibles.