Let

be a commutative ring and let

be the ring of polynomials in an indeterminate

, with coeﬃcients in

. Let

.

is said to be primitive if

. Prove that if

,

, then

is primitive iff

and

are primitive.

[This is Atiyah-Macdonald #1.2d]

I am particularly confused with

which I assume uses Gauss Lemma.

For the

, I have:

Suppose that

is primitive. Now suppose that

were not primitive.

Then

so there is a common factor to all of the terms. But then

would have a common factor as well. Thus

is primitive, and so must

be.