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 $\displaystyle d = p_1p_2 ... p_n $ . Since $\displaystyle p_1 $ is irreducible in R, the ideal $\displaystyle (p_1) $ is prime (cf Proposition 12, Section 8.3 - see attached) so by Proposition 2 above (see attached) the ideal $\displaystyle p_1R[x] $ is prime in R[x] and $\displaystyle (R/p_1R)[x] $ is an integral domain. ..."

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

(1) I cannot see why the ideal $\displaystyle (p_1) $ is a prime ideal. Certainly Proposition 12 states that "In a UFD a non-zero element is prime if and only if it is irreducible" so this means $\displaystyle p_1 $ is prime since we were given that it was irreducible. But does that make the principal ideal $\displaystyle (p_1) $ a prime ideal? I am not sure! Can anyone show rigorously that $\displaystyle (p_1) $ a prime ideal?

(2) Despite reading Proposition 12 in Section 8.3 I cannot see why the ideal $\displaystyle p_1R[x] $ is prime in R[x] and $\displaystyle (R/p_1R)[x] $ is an integral domain. ...". (Indeed, I am unsure that $\displaystyle p_1R[x] $ is an ideal!) Can anyone show explicitly and rigorously why this is true?

I would really appreciate clarification of the above matters.

Peter