Exercise 24, § 2

Let $A$ be a ring, commutative or not, and $\mathscr{F}$ the set of left ideals 1 of $A$ such that $l \cap m \neq 0$ for \emph{every} left ideal $m \neq 0$ of $A$.

In order that $l \in \mathscr{F}$, it is necessary and sufficient that, for every element $a \neq 0$ of $A$, there exist $b \in A$ such that $ba \neq 0$ and $ba \in l$. Every ideal $l$ of $\mathscr{F}$ contains the \emph{left socle} of $A$ (\emph{Algebra}, Chapter VIII, § 5, Exercise 9) and, if $A$ is Artinian, $\mathscr{F}$ consists of the ideals containing the left socle of $A$.

(a) Show that $\mathscr{F}$ is a topologizing set (Exercise 16). Deduce that the set $a$ of $a \in A$ such that $la = 0$ for some $l \in \mathscr{F}$ (union of the right annihilators of the $l \in \mathscr{F}$) is a two-sided ideal of $A$, which contains no idempotent. If $A$ is left Noetherian, show that $a$ is nilpotent (prove first that every element $a \in a$ is nilpotent, by considering the left annihilators of the $a^n$; then use Exercise 26 (b) of \emph{Algebra}, Chapter VIII, § 6).

Let $K$ be a commutative field and $B$ the ring of polynomials with coefficients in $K$ in indeterminates $Y$ and $X_i$, ($i \geq 1$). Let $b$ be the ideal of $B$ generated by the $X_i X_j$ ($i \# j$) and the $Y^{l-1} X_i$ and let $A$ be the quotient ring $B/b$; show that in $A$ the ideal $a$ defined above contains the class of $Y$, which is not nilpotent.

(b) Show that, for every left ideal $m$ of $A$, there exists a left ideal $n$ of $A$ such that $m \cap n = 0$ and $m + n \in \mathscr{O}$ (Take $n$ to be an ideal which is maximal amongst the left ideals $l$ such that $m \cap l = 0$).

(c) A ring $A$ is called (left) \emph{neat} if the two-sided ideal $a$ defined in (a) is reduced to 0. If this is so, the set $\mathscr{F}$ is idempotent (let $l, m, n$ be left ideals of $A$ such that $n \in \mathscr{F}$, $l \subset n$, $n/l$ being F-negligible, and $m \neq 0$. If $x \neq 0$ belongs to $m \cap n$, there exists $r \in \mathscr{F}$ such that $r.x \subset l \cap m$).