Who is ?
With this definition of , it's "easy" to show is a homomorphism with image (R/I)[x] and kernel I[x].
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.
yes, indeed, to apply the FIT, we need to know that φ is onto.
but suppose that f(x) is in (R/I)[x], that is f(x) has coefficients in R/I. recall that the NATURAL homomorphism R-->R/I which sends r-->r+I for each r in R IS onto. so every element b in R/I is a+I for SOME a in R.
so given f(x) = (a0+I) + (a1+I)x +...+ (an+I)xn in (R/I)[x], we clearly have for
g(x) = a0 + a1x +...+ anxn, φ(g(x)) = f(x), which shows that φ is surjective.