Can anyone tell me why the endomorphism ring of a finite-length module is artinian? Bonus points if you can do it without using the radical, semisimplicity, Fitting's lemma or anything fancy. If you have to or it makes the proof easier, that's OK too, but I have reason to believe that there's a simple proof (namely Dennis and Farb give this as an exercise in Chapter 0 of their book Noncommutative Algebra).



endomorphism ring of a finite-length module
Never mind. I was more successful with Google this time. It turns out that the statement is simply false. In the comment above I gave an example showing that the endomorphism ring need not be left Artinian. The following paper contains a (much less trivial) example showing that the endomorphism ring also need not be right Artinian:


"Ring of Endomorphisms of a Finite Length Module"
R. N. Gupta and Surjeet Singh
Proceedings of the American Mathematical Society, Vol. 94, No. 2 (Jun., 1985), pp. 198-200.


Incidentally, the module in that example appears to be a nontrivial self-extension of a simple module.


