Exercise 1, § 7

Let D be a field, V a right vector space over D, and A the ring End$_D(V)$. For any linear subspace W of V, we denote the set of elements $a$ of A satisfying $aW = 0$ (resp. $aV\subset W$) by $\mathfrak{a}(W)$ (resp. $\mathfrak{b}(W)$).

a) Prove that the mapping $W\mapsto \mathfrak{a}(W)$ (resp. $W\mapsto \mathfrak{b}(W)$) is a decreasing (resp. increasing) bijection from the set of linear subspaces of V to the set of left (resp. right) ideals of A generated by an idempotent.

b) The minimal left ideals of A are the ideals $\mathfrak{a}$(H), where H is a hyperplane in V. The minimal right ideals are the ideals $\mathfrak{b}$(D), where D is a line in V. Deduce the form of the left (resp. right) ideals of A of finite length.

c) Let W and $W'$ be two subspaces of V such that $W\not= 0$ and $W'\not= V$. Prove that the intersection $\mathfrak{a}(W)\cap \mathfrak{b}(W')$ is not reduced to 0.

d) The mapping $u\mapsto^tu$ defines a ring isomorphism from A to a subring B of End$_{D^o}(V^*)$. Show that the image of $\mathfrak{a}(W)$ (resp. $\mathfrak{b}(W)$) under this isomorphism is the trace on B of the ideal $\mathfrak{b}(W^o)$ (resp. $\mathfrak{a}(W^o)$), where $W^o$ denotes the orthogonal of W in $V^*$.

e) Let $\mathfrak{F}$ be the set of linear subspaces of V. For any left ideal $\mathfrak{a}$ of A, denote by $\mathfrak{F}(\mathfrak{a})$ the subset of $\mathfrak{F}$ consisting of the subspaces Ker $u$ where $u$ runs through $\mathfrak{a}$. The set $\mathfrak{F}(\mathfrak{a})$ is stable under intersection, and every linear subspace of V containing an element of $\mathfrak{F}(\mathfrak{a})$ belongs to $\mathfrak{F}(\mathfrak{a})$. Prove that the mapping $\mathfrak{a}\mapsto \mathfrak{F}(\mathfrak{a})$ is a bijection from the set of left ideals of A to the set of subsets of $\mathfrak{F}$ with these two properties. f ) Suppose that the dimension of V is infinite, and denote it by $\mathfrak{c}$. Let $\mathfrak{m}$ be a maximal left ideal of A that is not of the form $\mathfrak{a}(D)$. Prove that the intersection of the subspaces $W\in \mathfrak{F}(\mathfrak{m})$ is reduced to 0, that each of these subspaces is infinite-dimensional, and that there exist such subspaces of codimension $\mathfrak{c}$.