TITLE: Example of a ring without Invariant Basis Property
QUESTION [11 upvotes]: Let $A$ be a ring and consider the free modules $A^{\oplus n}$, $A^{\oplus k}$, with $n,k\in \mathbb{N}$. Can $A^{\oplus n}$ be isomorphic to $A^{\oplus k}$ if $k\neq n$?

Thanks in advance for the help.

REPLY [2 votes]: A ring $A$ is said to have (right) invariant basis property (IBP) if for any integers $s, t \geq 0$,
$$
A^{\oplus s} \cong A^{\oplus t} \Rightarrow s = t
$$
where the above isomorphism is as right $A$-modules.
Any non-zero commutative ring, any left Noetherian ring, and any semi-local ring satisfies IBP. The following link might be useful:  Cohn, P. M. - Some remarks on the invariant basis property. Topology 5 (1966) 215–228.