Is an injective endomorfism automatically bijective?
If you are talking about linear maps between finite-dimensional spaces, then yes. This is known as the Fredholm alternative: http://en.wikipedia.org/wiki/Fredholm_alternative
Thanks for confirming that intuition and for the pointer to Fredholm alternative, never heard of it.
Doesn't this follow from rank-nullity theorem?
for finitely generated modules over other rings this can fail, e.g. if Z is the integers, the injective endomorphism Z-->Z taking n to 3n is injective but not surjective. Interestingly however, a surjective endomorphism of a finitely generated module is always injective as well.
Yes. But the Fredholm alternative also hold in some infinite-dimensional cases. There, you can't use rank-nullity.
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