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## Homework Statement

Let I be an ideal of the commutative ring R, and let J = {y in R such that y^2 in I}

a) If R is the polynomial ring Q[x] and I is the principal ideal of R generated by x^4 + x^2, show that J is the principal ideal of R generated by x^3 + x

b) If R is a Principal Ideal Domain, show that J is an ideal of R

## Homework Equations

## The Attempt at a Solution

To solve for part a), I tried to write I = (x^4 + x^2) and (x^3 + x) explicitly and it turned out be long messy polynomial equations and i couldn't really show it rigorously that J = (x^3 + x) is true. For part b), one can take R to be Q[x] since Q[x] is a P.I.D. (since Q is a field), and by a), J must then be an ideal of R.

Any help on part a) particularly and b) would be very much appreciated.