Exercise 3, § 20

Let A be an Artinian ring and $\overline{A}$ the ring $A/\Re (A)$. We take a decomposition of A as a direct sum of indecomposable ideals $Ae_i(i\in I)$, where the $e_i$ are orthogonal idempotents (VIII, p. 179, Exercise 6, d)). We denote by $I =\cup_{\alpha\in R}I_{\alpha}$ the partition of I such that two elements $i, j$ of I belong to the same set $I_{\alpha}$ if and only if the modules $Ae_i$ and $Ae_j$ are isomorphic. The A-modules $\bigoplus_{i\in I_{\alpha}}\overline{A}\overline{e}_i$ are the isotypical components of A (VIII, p. 179, Exercises 6, d) and 7, a))

a) Prove that if the ring A is autoinjective (VIII, p. 53, Exercise 8), then there exists a permutation $\sigma$ of R such that every right ideal $e_iA$, for $i\in I_{\alpha}$, contains a unique minimal right ideal isomorphic to $\overline{e}_j\overline{A}$ for $j\in I_{\sigma(\alpha)}$ (observe that the right annihilator of an idempotent $e$ is $(1-e)A$). Deduce that the left socle $\mathfrak{s}$ of A is contained in its right socle $\mathfrak{t}$ (prove that every right ideal $e_i\mathfrak{s}$ is minimal, by proving that it is the annihilator of a maximal left ideal, cf. VIII, p. 178, Exercise 2). Conclude that we have $\mathfrak{s}=\mathfrak{t}$ and that every left ideal $Ae_i$, for $i\in I_{\sigma(\alpha)}$, contains a unique minimal left ideal isomorphic to $A\overline{e}_j$ for every $j\in I_{\alpha}$.

b) Conversely, if every left ideal $Ae_i$ contains a unique minimal left ideal and every right ideal $e_iA$ contains a unique minimal right ideal, then the ring A is autoinjective. (Use Exercise 7 of VIII, p. 179 to prove that every foot of $\mathfrak{s}$ is a minimal two-sided ideal. Deduce that $\mathfrak{t}$ contains $\mathfrak{s}$, and then that $\mathfrak{s}=\mathfrak{t}$. Finally, prove that condition $\beta$ ) of Exercise 8 of VIII, p. 53 is satisfied.)

c) Suppose that A is an algebra of finite degree over a commutative field and an autoinjective ring. Let $\alpha \in R,i\in I_{\alpha}$, and $j\in I_{\sigma(\alpha)}$. Prove that the linear representation of A associated with the A-module $Ae_j$ is isomorphic to the transpose of the linear representation of $A^o$ associated with the right A-module $e_iA$. (Let $U_i$ be the left submodule of $A^*$ that is the orthogonal of $(1-e_i)A$. Deduce from Exercise 7, d) of VIII, p. 179 that $U_i$ is isomorphic to a quotient module of $Ae_j$ and that we consequently have long $e_iA\leqslant$ long $e_jA$. Conclude that these lengths are equal, so $U_i$ is isomorphic to $Ae_j.)$ Use b) and Exercise 2, a) to prove a converse of this statement.