I'm trying to search for as many examples of non-IBN free modules as possible. But so far I can only find two examples. One is $End(V)$, where $V$ is a vector space of countable dimensions. Another is a similar one, mentioned in this question. They both involve an underlying vector space/module of infinite dimensions. Are there simpler examples of such modules? Because I can't seem to find any. Is the involvement of infinite dimensional objects implicit? Thanks.
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1Every (left) Noetherian ring has IBN. So, in a sense, a ring without IBN has to be big. – freakish Mar 17 '24 at 11:12