I have to prove the following
Let R be a unique factorization domain. Then the product of primitive polynomials in R[x] is primitive.
The author says the proof is the same of that for the case R = ℤ, just replace ℤ by R and prime by irreducible. The proof goes like this
Assume by contradiction that f(x)g(x) is not primitive. Then there exists a prime number p dividing all the coefficients of f(x)g(x). ...
My question is why have I to replace the notions of prime and irreducible element if in a unique factorization domain they coincide? Which is pretty much the same situation we have in $\mathbb{Z}$