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.


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