Purchase Solution

Ideals and Factor Rings : Prime Ideal

Not what you're looking for?

Ask Custom Question

Problem:
Let R be a commutative ring.
Show that every maximal ideal of R is prime and if R is finite, show that every prime ideal is maximal.
ALSO is every prime ideal of Z(integers) maximal? Why?

Purchase this Solution
Solution Summary

A proof involving a prime ideal is provided in the solution.

Solution Preview

Proof:
R is a commutative ring. Then M is an maximal ideal of R if and only if R/M is a field, P is a prime ideal if and only if R/P is an integral domain.
(1) If M is an maximal ideal, then R/M is a field. So if xy is in M, we consider x+M and y+M in R/M. Since R/M is a field, we have ...

Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Probability Quiz

Some questions on probability

View More Free Quizzes
Related BrainMass Solutions

  • Rings, Prime Ideals and Maximal Ideals

    72212 Rings, Prime Ideals and Maximal Ideals 1.Give a example of a commutative ring that has a maximal ideal that is not a prime ideal. 2. Prove that I=<2+2i> is not a prime ideal of Z[i]. How many elements are in Z[i]/I ?

  • Homomorphism of Commutative Rings, Prime Ideals, Maximal Ideals

    101848 Homomorphism of Commutative Rings, Prime Ideals, Maximal Ideals Let phi:R->S be a homomorphism of commutative rings a) Prove that if P is a prime ideal of S then either phi^-1(P)=R or phi^-1(P) is a prime ideal of R.

  • Ring Theory : Division Rings

    97572 Ring Theory : Division Rings Let R be a ring such that the only right ideals of R are (0) and R.

  • Ring and ideal proofs

    Now let I be an ideal of R, and consider the quotient map f : R --> R/I; then rad(I) = f^{-1} (nil(R/I)) Since nil(R/A) is the intersection of all the prime ideals of R/I, namely the ideals of R/I of the form P/I where P is a prime ideal

  • Ideals and Factor Rings : Locality

    17288 Ideals and Factor Rings : Locality Problem: A ring R is called a local ring if the set J(R) of nonunits in R forms an ideal. If p is a prime, show that Z(p) = {n/m belonging to Q | p does not divide m } is local.

  • Ideals and Factor Rings

    17279 Ideals and Factor Rings Problem: Note: | | is trying to denote a matrix If R = |S S| |0 S| and A = |0 S| |0 0| , S and ring, show that A is an ideal of R and describe the cosets in R/A Please see the attached file for

  • Commutative Rings, Homomorphisms and Ideals

    140602 Commutative Rings, Homomorphisms and Ideals Show that if R and S are commutative rings with 1, phi:R-->S is a homomorphism of R onto S, and I is an ideal of R, then phi[I]={phi(r): r included in I} is an ideal of S.

  • Problems Pertaining to Group Rings

    To see that the I_p are the only maximal ideals of Z[G] containing I, note that the I_n are the only ideals of Z[G] properly containing I. e. The maximal ideal from (d) which contains I and 2 is M = I_2.

  • Ideals and Factor Rings : Annihilators

    17293 Ideals and Factor Rings: Annihilators and Fermat's Theorem If R is a commutative ring, a polynomial f(x) in R[x] is said to annihilate R if f(a) = 0 for every a belonging to R Show that x^p - x annihilates Zp (Z is integers).

View More Related Solutions