Exercise 14, § 14

Let D be a field, Z its center, V a left vector space over D, and Ω the endomorphism ring of the abelian group V. Endow Ω with its natural $(D$, D)-bimodule structure. Let Γ be the automorphism group of D and Δ the subgroup of inner automorphisms (isomorphic to $D^*/Z^*$).

Let S be a subset of Ω consisting of semilinear mappings (relative to an element of Γ) and F the left D-linear subspace of Ω generated by S; it coincides with the right D-linear subspace generated by S.

a) For $\theta \in \Gamma /\Delta$, denote by $S_{\theta}$ the set of semilinear mappings from V to itself with respect to an automorphism of D belonging to $\theta$ and by $F_{\theta}$ the D-linear space of F generated by $S_{\theta}$. Prove that F is the direct sum of the subspaces $F_{\theta}$ for $\theta \in \Gamma /\Delta$. b) Prove that if $S_{\theta}$ is a free subset over Z, then it is also (left and right) free over D (reason as in the proof of Theorem 1 of V, §6, No. 1, p. 27).

c) Let E be a $(D$, D)-sub-bimodule of F. Prove that E is generated (over D) by semilinear mappings belonging to the $F_{\theta}$. (Let B be a basis of F contained in S. Prove that E is generated by the vectors $u\in E$ that have the following property: if $u=\sum^n_i\lambda_ib_i$ with $\lambda_i\not= 0$ and $b_i\in B$ for every $i$, then the intersection of E and the subspace of F generated by the $b_i$ is reduced to $Du$. Then deduce from a) that $u$ is semilinear.)