Exercise 8, § 10

Let A be an Artinian ring. The A-module $A_s$ is the direct sum of a finite family of indecomposable left ideals $(Ae_i)_{i\in I}$, where the $e_i$ are orthogonal idempotents (Exercise 6, d)).

a) Two A-modules of finite length M and N are said to be linked if a quotient of a Jordan–Hölder series of M is isomorphic to a quotient of a Jordan–Hölder series of N. Let R be the finest equivalence relation on the set of ideals $Ae_i$ for which two linked ideals are equivalent. The equivalence classes $\mathscr{B}_l(l\in L)$ for R are called blocks; we denote the sum of the ideals in block $\mathscr{B}_l$ by $\mathfrak{b}_l$. Prove that the ideals $\mathfrak{b}_l$ are the only indecomposable two-sided ideals of A.

(To see that $\mathfrak{b}_l$ is two-sided, observe that we have $Ae_ixe_j\subset Ae_j$ and that by Exercise 7, c), the relation $e_iAe_j\not= 0$ implies $Ae_i\equiv Ae_j$ (mod R). On the other hand, let $(\mathfrak{b}'_m)_{m\in M}$ be a finite family of two-sided ideals with direct sum A. Prove that each of the $\mathfrak{b}'_m$ is a sum of ideals $Ae_i$ and then that two linked ideals $Ae_r$ and $Ae_s$ belong to the same ideal $\mathfrak{b}'_m$ by observing that by Exercise 7, there exists an index $h$ such that $e_hAe_r$ and $e_hAe_s$ are $\not= 0.)$

b) Prove that the right ideals $e_iA$ are indecomposable (consider their images in $A/\mathfrak{R}(A)$).

c) Suppose that A is an algebra of finite degree over a commutative field K. We denote by $c_{ij}$ (resp. $d_{ij}$) the number of quotients of a Jordan–Hölder series of $Ae_j$ (resp. $e_jA$) isomorphic to $A\overline{e}_i$ (resp. $\overline{e}_iA$) and by $r_i$ the degree of the field $e_iAe_i/e_i\mathfrak{R}(A)e_i$ over K. Prove the equality $c_{ij}r_j=d_{ij}r_i$ (determine the dimension of $e_iAe_j$ over K, and use Exercise 7, c)).