# endomorphism ring of a finite-length module

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).

• Just to clarify: I think you want to say "right Artinian." In general the endomorphism ring does not have to be left Artinian. Namely, suppose $A$ is a ring that is left Artinian but not right Artinian (it is known that such rings exist). View $A$ as a module over itself using the left multiplication action. This module has finite length. The endomorphism ring is isomorphic to the opposite ring of $A$. That ring is right Artinian but not left Artinian. Commented Aug 1, 2010 at 0:26
• This is actually a cautionary tale that applies to all questions of the type "tell me why/when X is true": context (i.e. the origin of the question along with precise definitions or links to them) needs to be given. Commented Aug 1, 2010 at 0:48
• I think that the origin of the question was specified in the formulation, wasn't it? Commented Aug 1, 2010 at 1:34
• Perhaps this is a cautionary tale that applies to giving exercises in textbooks! Commented Aug 1, 2010 at 1:36
• @VP You might be able to view it with Google Books; however, there does not appear to be any essential difference in the statement. Commented Aug 1, 2010 at 17:25

• To completely put the matter to rest, I'll point out that in the example by Gupta and Singh, the ring is not right Artinian, but it is left Artinian. However, suppose we have rings $A$ and $B$ and modules $M$ over $A$ and $N$ over $B$ such that $R:=End_A(M)$ is not left Artinian and $S:=End_B(N)$ is not right Artinian. Then the product $C:=A\times B$ naturally acts on $M\times N$, and $End_C(M\times N)\cong R\times S$ is neither left nor right Artinian. Commented Aug 1, 2010 at 2:04