Exercise 9, § 7

Let D be a field, V a right vector space over D, and A a dense subring of the ring End$_D(V)$ (Exercise 3).

a) The subring A contains a nonzero endomorphism of finite rank if and only if the socle $\mathfrak{s}$ of A is nonzero (use Exercises 4, a) and 7, b) to prove that a minimal ideal of A is generated by an idempotent and then that this idempotent has rank 1). The socle of A then has a single foot (Exercise 4, a)).

b) From now on, assume $\mathfrak{s}\not= 0$. Let $\mathfrak{a}$ be a minimal left ideal of A. Prove that there exists an element $\ell$ of the dual $V^*$ of V such that $\mathfrak{a}$ consists of the endomorphisms $z\mapsto v \ell (z)$ with $v\in V$. Let $V'$ be the set of linear forms $\ell \in V^*$ such that the endomorphism $z\mapsto v \ell (z)$ belongs to A for every $v\in V$. Prove that $V'$ is a $(D,A)$-sub-bimodule of $V^*$ and that $\mathfrak{s}$ is the set of endomorphisms of V of finite rank whose kernel can be written as Ker $\ell_1\cap \cdots \cap$ Ker$\ell_n$ with $\ell_1, . . . , \ell_n$ in $V'$. We can only have $\mathfrak{s}= A$ if the vector space V is finite-dimensional and A = End$_D(V)$.

c) Let $\mathfrak{b}$ be a minimal right ideal of A. Prove that there exists a vector $v\in V$ such that $\mathfrak{b}$ consists of the endomorphisms $z\mapsto v \ell (z)$ with $\ell \in V'$. Deduce that the right socle of A is equal to its left socle $\mathfrak{s}$.

d) Prove that $\mathfrak{s}$ is, moreover, the smallest nonzero two-sided ideal of A.

e) Deduce from b) that a (left or right) Noetherian primitive ring with nonzero socle is simple (prove that V is necessarily finite-dimensional by considering the ideals contained in $\mathfrak{s}$).

f ) Let M be a finite-dimensional subspace of V. Prove that $\mathfrak{s}$ contains a projector $e_M$ of V with image M. (Let $N'\subset V'$ be a supplementary subspace of the orthogonal of M. Prove that there exist bases $(v_j)_{1\leqslant j\leqslant k}$ of M and $(\ell_i)_{1\leqslant i\leqslant k}$ of $N'$ such that $\ell_i(v_j) =\delta_{ij}$, and consider the endomorphism $x\mapsto \sum_iv_i\ell_i(x).$) Let N be the kernel of $e_M$. Prove that every endomorphism $u$ of V such that $u(M)\subset M$ and $u(N) = 0$ belongs to $\mathfrak{s}$.

g) Let J be a finite subset of A. Prove that there exists a decomposition $V = M\oplus N$, where M is a finite-dimensional subspace and N is the orthogonal of a finite subset of $V'$, such that the ring of endomorphisms $u$ of V such that $u(M)\subset M$ and $u(N) = 0$ is contained in A and contains J. (Let $M_1=\sum_{u\in J}$ Im$u$ and $N_1=\cap_{u\in J}$ Ker $u$, apply f) to a subspace M containing $M_1$ and such that $M + N_1= V$, and then take N contained in $N_1.)$