Exercise 19, § 5

Let A be a ring such that the A-module $A_s$ is the direct sum of a finite family of left ideals $(\mathfrak{I}_i)_{1\leqslant i\leqslant n}$ that are isomorphic as A-modules.

a) Prove that in A, there exists a family of elements $e_{i,j}(1\leqslant i\leqslant n, 1\leqslant j\leqslant n)$ such that $e_{i,j}e_{h,k}=\delta_{j,h}e_{i,k}$, where $\delta_{j,h}$ denotes the Kronecker delta function, and $\mathfrak{I}_i= Ae_{i,i}$ (cf. VIII, p. 39, Exercise 5 and I, §8, p. 173, Exercise 12). Conversely, if in A, there exists a family of $n^2$ elements $e_{i,j}$ such that $e_{i,j}e_{h,k}=\delta_{j,h}e_{i,k}$ and $1 =\sum^n_{i=1}e_{i,i}$, then the left ideals $Ae_{i,i}$ are isomorphic as A-modules (cf. VIII, p. 39, Exercise 5). Moreover, if B is the subring of A consisting of the elements that commute with all the $e_{i,j}$, then A is isomorphic to the matrix ring $\mathbf{M}_n(B)$ and B is isomorphic to the opposite ring of the commutant of each of the A-modules $Ae_{i,i}$ (use VIII, p. 78 and VIII, p. 65, Corollary).

b) Let M be an A-module. Prove that the $N_i=e_{i,i}M$ are B-modules and that M, viewed as a B-module, is the direct sum of the $N_i$; moreover, the B-modules $N_i$ are pairwise isomorphic (consider the mappings $x\mapsto e_{i,1}x$ and $y\mapsto e_{1,i}y$), and the annihilator of the $N_i$ is the intersection of B and the annihilator of M in A. Conversely, for every B-module N, define an A-module structure on the direct sum M of $n$ B-modules isomorphic to N, so that $e_{1,1}M$ is isomorphic to N.

c) To each B-submodule P of $N_1$, assign the A-submodule $\sum^n_{i=1}e_{i,1}P$ of M. Prove that this defines a bijective mapping from the set of B-submodules of $N_1$ to the set of A-submodules of M that is strictly increasing for the inclusion relation.