This is the problem I need to solve:
Let $A$ be a nonzero ring. Show that if $A^m ≅ A^n$, then $m = n$.
The book I got this problem from suggests using the following method to solve it:
Let $M$ be a maximal ideal of $A$ and let $f: A^m \rightarrow A^n$ be an isomorphism. Then $1\otimes f:(A/M) \otimes A^m \rightarrow (A/M) \otimes A^n$ is an isomorphism between vector spaces of dimensions $m$ and $n$ over the field $K = A/M$. Hence $m = n$.
The trouble is, I have no idea how to use this hint, mainly because I do not comprehend tensor products. We haven't gone over them in class due to some inclement weather closing it, and nothing I have read about them in the book or looking around on the internet, including a few answers to a question on this very website, makes sense to me. I just can't seem to get my head around what they are, how they're made from the modules they're made from, or what they're for.
Is there a way to solve this problem without using tensor products? And if there isn't, are there any proofs of other problems which use tensor products that I can read and maybe get an understanding of how to use this object in a proof?