Point 2 of my previous post on Gauss's Lemma got well lost in the 10 replies - so I am re-posting the remaining problem - which was problem (2)

Basically I am trying to understand the proof of Gauss's Lemma as given in Dummit and Foote Section 9.3 pages 303-304 (see attached)

On page 304, part way through the proof, D&F write:

"Assume d is not a unit (in R) and write d as a product of irreducibles in R, say . Since is irreducible in R, the ideal is prime (cf Proposition 12, Section 8.3 - see attached) so by Proposition 2 above (see attached) the ideal is prime in R[x] and is an integral domain. ..."

My remaining problem with the D&F statement above are as follows:

(2) Despite reading Proposition 12 in Section 8.3 I cannot see why the ideal is prime in R[x] and is an integral domain. ...". (Indeed, I am unsure that is an ideal!) Can anyone show explicitly and rigorously why this is true?

Would appreciate help.

Peter