Exercise 7, § 10

Let A be an Artinian ring, $\mathfrak{r}$ its radical, and $\overline{A}$ the ring $A/\mathfrak{r}$. Let $\mathfrak{l}$ be a nonnilpotent indecomposable left ideal; we have $\mathfrak{l}= Ae$, where $e$ is an idempotent in A.

a) Prove that the ideal $\mathfrak{n}=\mathfrak{l}\cap \mathfrak{r}$ is the unique maximal submodule of $\mathfrak{l}$. The A-module $\mathfrak{l}/\mathfrak{n}$ is simple and isomorphic to $A\overline{e}$, where $\overline{e}$ is the image of $e$ in A.

b) Let M be an A-module. Then $(Ae)M$ is a submodule of M, and $e((Ae)M)=eM$ is an $eAe$-module. If N is an A-submodule of M, then the $eAe$-modules $e(M/N)$ and $eM/eN$ are isomorphic. If the A-module M is simple, then either we have $eM = 0$, or M is isomorphic to $A\overline{e}$ and $eM$ is a simple $eAe$-module.

c) Prove that the length of the $eAe$-module $eM$ is equal to number of quotients of a Jordan–Hölder series of the A-module M isomorphic to $A\overline{e}$.

d) Suppose that M has finite length and admits a unique maximal submodule N and that $M/N$ is isomorphic to $\mathfrak{l}/\mathfrak{n}$. Prove that M is isomorphic to a quotient of $Ae$ (observe that an element $x$ of $eM$ that does not belong to N generates M).