Purchase Solution

Polynomial Rings : Prove that f (x) is a zero - divisor in A[x] IF AND ONLY IF (iff) 7c ! A, c ! 0 " cf (x) = 0.

Not what you're looking for?

Ask Custom Question

Let A be a commutative ring with identity 1, and let A[x] be the ring of polynomials with coefficients in A.
Let f (x) = a0+ a1x + ... + anxn! A[x] .
Prove that f (x) is a zero - divisor in A[x] IF AND ONLY IF (iff) 7c ! A, c ! 0 " cf (x) = 0.

Please see the attached file for the fully formatted problems.

Attachments
Purchase this Solution

Solution Summary

Commutative rings and a zero-divisor are investigated. The solution is detailed and well presented.

Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

Recall: let R is a ring. Let a  0  R. Then we say a is a zero divisor if there exist b  0 R such that a.b = 0

Let f(x) = a0 + a1x + ...+ anxn. If a0 is a zero divisor of A, then it is a zero divisor of A[x] such that a0b0 =0 for some b0  A. Let g(x) = b0 + b1x+ ... + bmxm  A[x]. Then we have
a0.g(x) = ...

Purchase this Solution


Free BrainMass Quizzes
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.

Probability Quiz

Some questions on probability

Graphs and Functions

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

Multiplying Complex Numbers

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

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.