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.
let and let first see that by the definition of the coefficients if then for
some so if is primitive, clearly both and must be primitive too.
conversely, suppose and are primitive but is not primitive. so , for some maximal ideal of now for any let so thus:
but is an integral domain. thus either or which means either or contradiction!
Alternatively:
=> contrapositive. wlog suppose f is not primitive. let I be the proper ideal generated by the coefficients of f. It is easily seen that all the coefficients of fg lie in I so that fg is not primitive.
Other direction requires a rigorous argument like above.