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I am not so very familiar with the mathematics of modules over rings. I heard of them as a generalization of vector spaces with a ring instead of a field. I also heard of them as providing many pathological examples when thinking in the familiar term of vector spaces (there was an MO thread discussing examples, but I cannot find it). E.g. I learned that an $R$-module (over some ring $R$) does not necessarily have a basis, even though it has independent and generating sets. My immediate next question then was:
Question: Can an $R$-module have bases of different cardinality?
I am especially interested in different finite cardinalities. Can you give me an example of such a module? Or is it that the dimension of an $R$-module (if at all) can be still defined uniquely?