Let $R$ be a ring with unity $1$ and artinian. Consider $R$ as right $R$-module $R_R$. Since it is artinian we have a finite decomposition of $R$ into right ideals:

$$R_R = A_1 \oplus ... \oplus A_n$$ where each $A_i$ is an indecomposable right ideal generated by an idempotent $e \in R$, that is $A_i = e_iR$ for every $i=1,...,n$.

I want to prove that each $e_iR$ is a local right $R$-module. By a local module I mean cyclic, non-zero and has a unique maximal proper submodule.

Do you have any idea?

  • $\begingroup$ The hypotheses are not enough to guarantee it: $e=1$ is idempotent, but $eR$ is not necessarily local. Maybe $A_1,\dots, A_n$ are indecomposable? $\endgroup$ – egreg Jun 29 '18 at 8:35
  • $\begingroup$ You're right, i've edited. $\endgroup$ – bozcan Jun 29 '18 at 8:46

Let $J$ be the Jacobson radical of $R$. Then $R/J$ is a semisimple artin ring which can be written as a direct sum of $n$ simple modules. Actually each $A_i/A_iJ$ is simple $R$-module and $rad(A_i)=A_iJ$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.