Q1. (a) Find all the prime ideals and maximal ideals of the quotient ring Z/20Z. Are the set of prime ideals and the set maximal ideals of Z/20Z the same?

(b) Let n ≥ 2. Is it true that every prime ideal of the quotient ring Z/nZ is a maximal ideal? Prove it if it is true, give a counterexample if it is false.

Q2. Let P be an ideal of a ring R. and let P[x] denote the set of all polynomials in R[x] with coefficients in P.

(a) Prove that P is a prime ideal of R if and only if P[x] is a prime ideal of R[x].

(b) If P is a maximal ideal of R, is it true that P[x] is a maximal ideal of R[x]? Justify your answer.

Q3. Let R be a ring. Let N be the set of all nilpotent elements in R.

(a) Prove that N is an ideal of R. (This ideal is called the nilradical of R.) [You may assume that the binomial theoremholds in any ring R, where a, b ∈ R and n is a positive integer.]

(b) Prove that 0 + N is the only nilpotent element in the quotient ring R/N.

Q4. Let k be a field. Prove that the ideal (x, y) generated by x and y in the polynomial ring k[x, y] is a maximal ideal.

Q5. Let R be a ring, and let Spec(R) denote the set of all prime ideals of R. Use Zorn's Lemma to prove that Spec(R) has a minimal element with respect to set inclusion; that is, there exists P ∈ Spec(R) such that whenever Q ⊆ P with Q ∈ Spec(R), then Q = P.

Q6. Let Z[i] = {a + bi: a, b ∈ Z}, where i² = -1. It is given that Z[i] is a ring. Prove that the principal ideal (3) generated by 3 is a maximal ideal of Z[i].

Provide full solutions to the assignment questions.