Question: Let φ:R→S be a homomorphism of commutative rings. (a) Prove that if P is a prime ideal of S then either φ−1(P)=R or φ−1(P) is a prime ideal of R. Apply this to the special case …

Question: Recall that every maximal ideal is a prime ideal. The converse, however, is not true. Showthat {0} is a prime, but not maximal, ideal of Z.

Question: #7. P256 Let R be a commutative ring with 1. Prove that the principal ideal generated by x in the polynomial ring R[x] (real coefficients) is a prime ideal if and only if R is an integral …

To show that the characteristic of divides , consider the ideal . Then, express in terms of its equivalence class modulo . (a) Show that the characteristic of divides a2 + b2 Since Z[i] / …

Prove that every prime ideal of R is a maximal ideal. Assume R is commutative and for each there is an integer n > 1 (depending on a) such that a n =a. Show transcribed image text

First note that if F is field , then F[x] is a Euclidean domain and thus a Principal ideal domain.So it suffices to show that in a PID , every nontrivial prime ideal is maximal.Let I be a nontrivial prim …

Let f: R S be an epimorphism of rings with kernel K. (a) If P is a prime ideal in R that contains K, then f(P) is a prime ideal in S (see Exercise 13]. (b) If Q is a prime ideal in S, then f-¹(Q) is a …

Then show that Z × {0} is a prime ideal of Z × Z Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on.

Question: 1. Find all the prime ideals and all maximal ideals of Z12. 2. Let R be a finite commutative ring with identity.

Give an example of a commutative ring that has a maximal ideal that is not a prime ideal. Solve by considering R = {0,2,4,6} under addition modulo 8, multiplication modulo 8. You can assume …