Math 171: Abstract Algebra, Fall 2014- Assignment 9
1. Decide which of the following are ideals of the given ring.
(a) the set of all polynomials whose constant term is a multiple of 3, in the ring Z[x].
(b) the polynomials in Z[x] whose exponents all have even powers.
(c) the set of polynomials f(x, y, z) that vanish on a fixed subset S ⊆ R3, in the ring R[x, y, z]. (i.e. the set { f ∈ R[x, y, z] | f(a, b, c) = 0 for all (a, b, c) ∈ S}).
(d) the set of polynomials p(x) such that p'(0) = 0, where p'(x) is the usual first derivative of p(x) with respect to x, in the ring Z[x].
2. In this problem we discuss the set R/I and the construction of a ring from this set when I is an ideal. Recall that for x, y ∈ R, we have that the set product (x + I)(y + I) sits inside the set xy + I so the coset xy + I is a natural candidate for the product of x + I and y + I in R/I (though it may or may not be the case that as a set, (x + I)(y + I) = xy + I). Now define operations ⊕, ⊗ on R/I by
(x + I) ⊕ (y + I) = (x + y) + I
(x + I) ⊗ (y + I) = xy + I
Using the axioms for rings, show that R/I with addition ⊕ and multiplication ⊗ forms a ring. Note: You need to show that these operations are well-defined (i.e. they do not depend on a choice of coset representative).
3. Let R and S be nonzero rings with identity and denote their respective identities by 1R and 1S. Let φ: R → S be a nonzero homomorphism.
(a) Prove that if φ(1R) ≠ 1S then φ(1R) is a zero divisor of S. Deduce that if S is an integral domain then φ(1R) = 1S.
(b) Prove that if φ(1R) = 1S then φ(u) is a unit for any unit u in R and φ(u)-1 = φ(u-1).
4. Let ?: R → S be a ring homomorphism.
(a) Prove that if J is an ideal of S then ?-1 (J) is an ideal of R. Apply this to the special case when R is a subring of S and ? is the inclusion homomorphism (i.e. ?: R → S is given by ?(r) = r) to deduce that if J is an ideal of S then J ∩ R is an ideal of R.
(b) Prove that if ? is surjective and I is an ideal of R then ?(I) is an ideal of S. Give an example where this fails if ? is not surjective.
5. Let p ≥ 2 be a prime, and let R = Z/pZ[x] be the ring of polynomials with coefficients in Z/pZ. If f(x) ∈ R, let I = (f(x)) be the ideal consisting of the polynomials r(x)f(x) where r(x) ∈ R.
(a) If f(x) has degree d, what is the size of the quotient ring R/I?
(b) If p = 2 and f(x) = x2 + x + 1, show that R/I is a field.
(c) If p = 3 and f(x) = x2 + x + 1, show that R/I is not a field.
6. Let R = Z[√2]. Let I = (3) := {3x| x ∈ R}.
(a) Find representatives of the equivalence classes of R/I. How many are there?
(b) Find inverses for all non-zero equivalence classes of R/I.