Let $R$ be a nonzero ring and $M$ be a free module on $R$ and here are theorems I know.

If $M$ has a infinite basis, then rank of $M$ is well-defined

If $R$ has IBN property, then rank of $M$ is well-defined.

This means that for any $R$ and $M$, infinite-rank of $M$ can be well-defined. So well-definedness of rank only depends on finite-rank cases.

However, in general, the rank of a module is defined only for rings with IBN property even though one can extend the domain of $rank$ for any infinite-dimensional module. Why?

From my experience, in a vector space, things do not go fluently when it is infinite-dimensional, but with additional structures, things work fluently again. (e.g. Banach space and Hilbert space)

I'm curious whether the reason of not defining ranks over rings without IBN property is that there is currently no use of modules over these rings. Is it? Or is it because main parts of ring families such as division ring and commutative ring have IBN property?


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