Is there a free $R$-module $M$ such that $M$ has invariant basis number (IBN), while $R$ does not have the IBN property?
My use of "invariant basis number" here (for modules) doesn't seem to be standard, but I just mean $M$ has a unique, well-defined rank. To rephrase my question, can there exist a free $R$-module $M$ such that
- Every basis of $M$ has the same cardinality.
- There exists a free $R$-module that has bases of different cardinalities.
Just curious, as I'm only first learning about IBN and don't know many examples yet.
(I mention in passing the zero module, which is free and with a unique number of generators for any ring, but it is not really in the spirit of the question.)
Using Leavitt path algebras, it is possible to construct a ring such that $R^2\cong R^3$, and hence $R^2\cong R^n$ for all $n\geq 2$, but $R\ncong R^2$. (Here is another post about it.)
In that case, every cyclic free $R$ module has a unique rank, but free modules with more than one generator do not.
You're right that one shouldn't probably call this "invariant basis number for a module" but rather say that it's a free module "whose bases have a unique, finite size" or something like what you said.