In Dummit and Foote Ch 9 Polynomial Rings, Proposition 2 (page 296 - see attached) reads as follows:
Proposition 2: Let I be an ideal of the ring R and let (I) = I[x] denote the ideal of R[x] generated by I (the set of polynomials with co-efficients in I)
In particular, if I is a prime ideal of R then I[x] is a prime ideal of R[x]
I am concerned with the proof that .
In this proof (i think) we need the First Isomorphism Theorem. Dmmit and Foote give this theorem as part of Theorem 7 in Ch 7 on page 243 (see attached)
They state the First Isomorphism Theorem as follows: (see attached)
If is a homomorphism of rings, then the kernel of is an ideal of R, the image of is a subring of S, and is isomorphic as a ring to
I follow the proof of down to the following statement (see attached page 296 D&F)
... ... , which is to say that ker = I[x] = (I), proving the first part of the proposition.
BUT surely we need to now use the First Isomorphism Theorem to show that we have an isomorphism ( I suspect D&F are assuming we know this bit ... )
However, by the FIT, given that ker = I[x] we have that
So we require
But how do we know that the image of R[x] under is (R/I)[x] and not a subset of (R/I)[x]
Would appreciate some help in clarifying this matter.