[csense]

> injections on finite sets are also surjections

Not necessarily [1]. I think you're missing an assumption there.

[1] https://en.wikipedia.org/wiki/Injective_function#/media/File...

[MaxRegret]

In this case, multiplication by any nonzero fixed element of the ring is an injection from the ring to itself. Any injection from a finite set to itself is indeed a surjection (and so also a bijection).

[susam]

The intended point, I believe, is the fact that any injective function from a finite set to itself is also surjective.