Irreducible but not prime

I need an example of a ring that is irreducible but not prime.

there is this example of such a ring in the book saying in Z[-(root(3)] where

4= 2*2

but also 4= (1+root(-3))*(1-root(-3))

okay I get it that and I know 1+root(-3) doesn't divide 2 hence is is not prime. but also if it was reducible then we should have units

since the definition of irreducible is that a=b*c then b or c is unit. what is the unit of this ring !!!!?

(ps: if one day I understand everything in abstract algebra, surely I will rewrite the book)