Exercise 16, § 14

Keep the notation of Exercise 15, and suppose, moreover, that V is finite-dimensional over D. A subring B of A is called weakly Galois (resp. Galois) in A if its commutant $B'$ in Ω is generated as a (left or right) D-vector space by semilinear mappings (resp. bijections) from V to itself. Denote by DB the subring of Ω generated by D and B.

a) Prove that if B is simple and weakly Galois, then V is a semisimple module of finite length over DB. (Let W be an isotypical component of the B-module V and S a simple DB-submodule of W. Let Σ be the set of semilinear mappings $v$ belonging to $B'$. Observe that $v(S)$ is an DB-module for $v\in \Sigma$ and deduce from Corollary 3 of VIII, p. 56 that W is contained in the sum of the $v(S)$ for $v\in \Sigma .$)

b) Suppose that the ring B is simple and weakly Galois and that its commutant $B'_0$ in A is simple. Prove that V is an isotypical DB-module (observe that $B'_0$ is the commutant of DB in Ω, and use Proposition 5 of VIII, p. 85). Deduce that B is Galois. (Let $v$ be a nonzero element of Σ and $V =\bigoplus^n_{i=1}V_i$ a decomposition of V as a direct sum of isomorphic simple DB-modules such that $W_1=v(V_1)$ is nonzero. Observe that there exist simple DB-modules $W_i$ (for $2\leqslant i\leqslant n$) isomorphic to $W_1$ such that $V =\bigoplus^n_{i=1}W_i$, and deduce a semilinear bijection $w\in B'$ that coincides with $v$ on $V_1$ and for which $vw^{-1}$ belongs to $B'_0$. Conclude by observing that every element of a simple ring is a sum of invertible elements.)