# Ideals and Factor Rings : Prime Ideal

Not what you're looking for?

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