I am reading Dummit and Foote Chapter 9 on Polynomial Rings and am trying to get a good understanding of Propostion 2 (see Attachment - page 296) which states:

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Let I be an ideal of the ring R and let (I) = I[x] denote the ideal of R[x] generated by I (set of polynomials with co-efficients in I). Then

R[x]/I[x] $\displaystyle \cong $ (R/I) [x]

In particular, if I is a prime ideal of R then (I) is a prime ideal of R[x]

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I decided to generate a simple example using R= $\displaystyle \mathbb{Z}$ and I = 2$\displaystyle \mathbb{Z}$

Then $\displaystyle \mathbb{Z}$ = { ..., -2, -1, 0, 1, 2, 3, ... } and 2$\displaystyle \mathbb{Z}$ = { ..., -4, -2, 0, 2, 4, 6 .... }

also $\displaystyle \mathbb{Z}$/2$\displaystyle \mathbb{Z}$ = { $\displaystyle \overline{0}, \overline{1}$ }

Then it appears to me that R[x] = $\displaystyle \mathbb{Z}$ [x] is the set of all polynomials with integer coefficients and I[x] is the set of polynomials with even integers as coefficients

Now how do I formally express R[x], I[x] and $\displaystyle \mathbb{Z}$/2$\displaystyle \mathbb{Z}$ formally and algebraicly?? Is my text above OK?

It seem that $\displaystyle \mathbb{Z}$ [x] /2$\displaystyle \mathbb{Z}$ [x] would have two elements - one which was all the polynomials with even co-efficients and one which contains all the polynomials with odd integer co-efficients - but again - how do I express this in formal algebraic symbolism

Further, returning to Propostion 2 above

R/I = $\displaystyle \mathbb{Z}$/2$\displaystyle \mathbb{Z}$ = { $\displaystyle \overline{0}, \overline{1}$ } and so (R/I)[x] appears to have two elements - one the set of polynomials with coefficients in $\displaystyle \overline{0}$ and one with coefficients in [TEX]\overline{1}TEX] which seems to be correct.

Is my example and reasoning correct? Would appreciate an assurance from someone that all is OK.

How would my exampole ber expressed in more formal algebraic symbolism?

Note: I was also somewhat thrown by D&F's use of the symbolism (I) for I[x]. Previously (see attachment on Properties of Ideals, D&F ch 7 page 251) the symbol (I) was used to denote the smallest idea of R containing I which D&F point out is the set of all finite sums of elements of the form ra with r $\displaystyle \in $ R and a $\displaystyle \in $ I ie sums such as $\displaystyle r_1 a_1 + r_2 a_2 + ... + r_n a_n$. Isn't the use of (I) for I[x] somewhat inconsistent with the use of the symbolism just described.

Peter