Endomorphisms of modules satisfying chain conditions and counterexamples.

Solution 1:

The keywords for these phenomena are that a module is Hopfian if surjective endomorphisms are isomorphisms, and coHopfian if injective endomorphisms are isomorphisms.

It's indeed true that Noetherian modules are Hopfian and Artinian modules are coHopfian.

The strategies more or less presents themselves:

Suppose $M$ is Artinian and that $f$ is an injective endomorphism of $M$. Look at the chain of images $\operatorname{Im}(f)\supseteq \operatorname{Im}(f^2)\supseteq \operatorname{Im}(f^3)\supseteq\cdots$ and consider what would happen if this chain stabilized at some point.

Suppose $M$ is Noetherian and that $f$ is a surjective endomorphism of $M$. Look at the chain of images $\ker(f)\subseteq \ker(f^2)\subseteq \ker(f^3)\subseteq\cdots$ and consider what would happen if this chain stabilized at some point.

If you get stuck on one, you might check the other to see if you overlooked some idea that dualizes to the other.

Solution 2:

Fitting's Lemma is what you are looking for.

For the counterexamples, consider the ring $\mathbb{Z}[x_1,x_2,x_3,\dots].$ The ring endomorphism that sends $x_i \to x_{i+1}$ is injective but not surjective (the image does not contain $x_1$). The endomorphism which sends $x_i \to x_{i-1}$ for $i\geq 2$ and $x_1\to 1$ is surjective but not injective.