6
$\begingroup$

Suppose that $R$ is a commutative ring. I'm wondering if the space $R^{\mathbb N}$ is a free $R$ module.

I know how to prove that it is not a free $R$ module in the case of $R = \mathbb Z$. But the proof I know/could think of, uses facts specific of $\mathbb Z$ such as it being an Euclidean ring. So I was unable to recycle this idea for a generic $R$. Also I know that if $R$ is a field, then every $R$-module is free. So in particular $R^{\Bbb N}$ is a free $R$ module. So the question is,

What happens when $R$ is a commutative ring but not a division ring?

As an example $R=\Bbb Z_n$ if it is too broad a question.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.