This question is from the book A course in Homological algebra by Hilton and Stammbach
Let $V$ be a vector space of countable dimension over the field $K$. Let $\Lambda=\rm{Hom}_K(V,V)$. show that, as $K$ vector spaces $V$, is isomorphic to $V\oplus V$. We therefore obtain $$\Lambda=\rm{Hom}_K(V,V)\cong \rm{Hom}_K(V\oplus V,V)\cong\rm{Hom}_K(V,V)\oplus \rm{Hom}_K(V,V)$$
Conclude that, in general, the free module on a set of $n$ elements may be isomorphic to the free module on a set of $m$ elements, with $n\neq m$.
Let $V$ be a vector space of countably infinite dimension. We then have a basis for $V$ as $\{e_n: n\in \mathbb{N}\}$. This gives a basis for $V\oplus V$ as $\{(e_n,e_m):n,m\in \mathbb{N}\}$. As we have a bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$, we have an isomorphism between $V$ and $V\oplus V$.
By properties of homomorphism ring, we have $$\Lambda=\rm{Hom}_K(V,V)\cong\rm{Hom}_K(V\oplus V,V)\cong\rm{Hom}_K(V,V)\oplus \rm{Hom}_K(V,V)$$
Now, $\Lambda$ is free as $\Lambda$ module of dimension $1$. We have $\Lambda\oplus \Lambda$ free as $\Lambda$ module of dimension $2$.
Nevertheless we have isomorphism between $\Lambda$ and $\Lambda\oplus \Lambda$.
Thus, the free module on a set of $n$ elements may be isomorphic to the free module on a set of $m$ elements, with $n\neq m$.
Please let me know if this justification is correct.
Provide some more examples of this kind.