Purchase Solution

Module homomorphism proof

Not what you're looking for?

Ask Custom Question

Let φ: Z "circle +" Z → Z "circle +" Z be a module homomorphism (of Z-modules). Show that if φ is surjective, it must be injective. Give an example to show that the converse is false ─ a difference between free Z-modules and vector spaces. (You may, of course, think of φ as a 2 x 2 matrix with integer entries.)

Please see attached document.

Purchase this Solution

Solution Summary

This provides a proof that if a module homomorphism is surjective it is injective.

Solution Preview

Please see the attachment.

Proof:
We consider two special values , , where . Then I claim that for any , we have .
Since is a module homomorphism, then we have

.
Thus , for any .
Now we show that if is a surjective, then ...

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.

Probability Quiz

Some questions on probability

Exponential Expressions

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

Solving quadratic inequalities

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

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.