Exercise 10, § 9

Let A be a ring.

a) Let Z be the center of A. Prove that we have $Z\cap \mathfrak{R}(A)\subset \mathfrak{R}(Z)$ (cf. Exercise 15, b) below for an example where the inclusion is strict).

b) Prove that $\mathfrak{R}(A)$ contains no nonzero idempotents.

c) Prove that the intersection of $\mathfrak{R}(A)$ and the left socle $\mathfrak{s}$ of A (VIII, p. 130, Exercise 8) is a two-sided ideal with square zero and that it is the intersection of $\mathfrak{s}$ and its left annihilator (use Exercises 7 and 8, a) of VIII, p. 130).

d) Let $e$ be an idempotent in A. Prove that the radical of the ring $eAe$ is $e\mathfrak{R}(A)e$ (observe that for $a\in eAe$, we have $(1-a)(1-e) = (1-e)(1-a) = 1-e$).

e) Prove that the radical of the ring $\mathbf{M}_n(A)$ is the ideal $\mathbf{M}_n(\mathfrak{R}(A))$ (VIII, p. 114, Example 2).