Math 366–001 HW 6, Spring 2015

Math 366–001 HW 6, Spring 2015
This assignment is due Friday, April 10 in class. Feel free to work together, but be
sure to write up your own solutions.
As for writing it up, please write legibly on your own paper, including as much justification
as seems necessary to get the point across.
1. (2 pts) If A and B are ideals of some ring, prove that AB ⊂ A ∩ B.
2. (2 pts) If I is an ideal of a ring R and I contains a unit, show that I = R.
3. (1 pt) Find the two zeros of x2 + 3x + 2 in R and the four zeros of it in Z6 .
4. (1 pt) List all polynomials of degree 2 in Z3 [x].
5. (1 pt) Determine the quotient and remainder when dividing x3 + 2x + 4 by 3x + 2 in
Z5 [x].
6. (3 pts) Suppose you know an isomorphism φ : R → S. Use φ to construct an isomorphism ψ : R[x] → S[x]. In particular, define the image of a polynomial
p(x) =
ai xi ∈ R[x]
under ψ and show that
ψ(p(x) + q(x)) = ψ(p(x)) + ψ(q(x))
ψ(p(x)q(x)) = ψ(p(x))ψ(q(x)).