Exercise 2, § 7

Let D be a field, V a right vector space of infinite dimension $\mathfrak{c}$ over D, and A the ring End$_D(V)$.

a) Let $u$ and $v$ be endomorphisms of V such that rg($u$)$\leqslant$ rg($v$). Prove that there exist elements $a$ and $b$ of A such that $u=avb$.

b) For any infinite cardinal $\mathfrak{n}\leqslant \mathfrak{c}$, denote the set of endomorphisms of V of rank $<\mathfrak{n}$ by $\mathfrak{A}_{\mathfrak{n}}$. Prove that $\mathfrak{A}_{\mathfrak{n}}$ is a two-sided ideal of A and that every two-sided ideal of A distinct from $\{0\}$ and A is one of the $\mathfrak{A}_{\mathfrak{n}}$ (use a)). In particular, the ideal of endomorphisms of finite rank is the smallest nonzero two-sided ideal of A, and the ideal $\mathfrak{A}_{\mathfrak{c}}=\mathfrak{T}$ of endomorphisms of rank $<\mathfrak{c}$ is the greatest two-sided ideal of A distinct from A.

c) Let B be a basis of V and $\mathfrak{U}$ an ultrafilter on B that is finer than the filter $\mathfrak{G}$ of complements of subsets of B of cardinal $<\mathfrak{c}$. For any set U in $\mathfrak{U}$, denote by $e_U$ the projector of V with kernel generated by the vectors of U and image generated by the vectors of B - U. Let $\mathfrak{a}_{\mathfrak{U}}$ be the left ideal of A generated by the $e_U$, where U runs through $\mathfrak{U}$. Prove that the ideal $\mathfrak{T}+\mathfrak{a}_{\mathfrak{U}}$ is distinct from A. Let $\mathfrak{V}$ be another ultrafilter on B containing $\mathfrak{G}$ and distinct from $\mathfrak{U}$. Prove that we have $\mathfrak{a}_{\mathfrak{U}}+\mathfrak{a}_{\mathfrak{V}}= A$. d) Prove that the cardinal of the set of ultrafilters on B containing $\mathfrak{G}$ is $2^{2^{\mathfrak{c}}}$. (Use the arguments of Exercises 6 of Gen. Top., I, §8, p. 135 and 5 of Gen. Top., I, §4, p. 126. Observe that in the latter exercise, the trace on F of every nonempty open subset of X has the same cardinal as F.)

e) Suppose Card(D) $\leqslant 2^{\mathfrak{c}}$. Deduce from d) that the set of classes of simple $(A/\mathfrak{T})$-modules has cardinal $2^{2^{\mathfrak{c}}}$ (cf. VIII, p. 52, Exercise 5, d)).

f ) Deduce from the above that the ring A is primitive but not quasi-simple and that the ring $A/\mathfrak{T}$ is quasi-simple but not simple (VIII, p. 120, Remark 2).