Exercise 8, § 7

Let A be a ring. The left socle (or simply socle)$\mathfrak{s}$ of A is the socle of the A-module $A_s$, that is (VIII, p. 65), the sum of the minimal left ideals of A. It is a semisimple A-module, and its isotypical components are called its feet. Let $\mathfrak{a}$ be a foot of $\mathfrak{s}$.

a) Prove that $\mathfrak{a}$ is a two-sided ideal and that the sum of the minimal ideals of square zero (Exercise 7) contained in $\mathfrak{a}$ is a two-sided ideal, namely the intersection of $\mathfrak{a}$ and its annihilator.

b) Suppose that $\mathfrak{a}$ contains a minimal ideal with nonzero square. Prove that there are no nonzero elements $x$ of $\mathfrak{a}$ such that $\mathfrak{a}x= 0$ (use Exercise 7, c)).

c) Under the assumption of b), prove that every left ideal of the pseudoring $\mathfrak{a}$ (I, §8, No. 1, p. 98) is an ideal of A (observe that the minimal ideals of $\mathfrak{a}$ are the minimal ideals of A contained in $\mathfrak{a}$).

d) Suppose that the socle $\mathfrak{s}$ of A does not contain any minimal left ideals of square zero. Prove that every left ideal $\mathfrak{b}$ of A that has finite length as an A-module is contained in $\mathfrak{s}$ (use induction on the length of $\mathfrak{b}$ by observing that if $e$ is an idempotent in $\mathfrak{b}$, then $\mathfrak{b}e$ is a direct factor of $\mathfrak{b}$).