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**[SOLVED] Algebra question - rings and ideals**

## Homework Statement

Let R be a (nonzero) commutative ring with identity and I be an ideal of I. Denote (I) the ideal of R[x] generated by I. The book says that (I) is the set of polynomials with coefficients in I. Why is that?

## The Attempt at a Solution

Call A the set of polynomials with coefficients in I.

R is commutative and so is R[x], therefor (I) is simply given by

(I) = {a*p(x): a is in I and p(x) in R[x]}

So clearly (I) is a subset of A.

But for the other inclusion, given b_0+...+b_nx^n in A, we need to find an element a in I and a set {a_0,...,a_n} in R such that a*a_i = b_i for all i=1,...,n.

How is this achieved??