In the case of finite-dimensional vector spaces, an endomorphism is injective if and only if it is surjective. In the case of finitely generated modules over a commutative ring, if an endomorphism is surjective, then it is injective. I am wondering about the converse, which i suspect is not true. I am interested in counterexamples that give insight into why an injective endomorphism of a finite module is not necessarily surjective.
Edit: at several occasions in abstract algebra injectivity and surjectivity appear to be dual notions. At other cases this duality breaks down. The case i am referring to seems to be one of these and i am interested in understanding what it is that makes one notion (surjectivity) "more difficult" to achieve than the other (injectivity).