Exercise 10, § 10

For a ring A, denote the set of pseudosubrings of A whose elements are all nilpotent by $\mathscr{N}(A)$ and the set of maximal elements of $\mathscr{N}(A)$ by $\mathscr{N}_{max}(A)$.

a) Let D be a field and $n$ an integer. Prove that $\mathscr{N}_{max}(\mathbf{M}_n(D))$ consists of the pseudorings $gTg^{-1}$, where $g$ is an invertible element of $\mathbf{M}_n(D)$ and T is the set of matrices $(a_{ij})\in \mathbf{M}_n(D)$ such that $a_{ij}= 0$ for $i\geqslant j$ (use Exercise 2 of VIII, p. 14). b) Let A be an Artinian ring and $\pi : A\rightarrow A/\mathfrak{R}(A)$ the canonical mapping. Show that the mapping $P\mapsto \pi (P)$ defines a bijection from $\mathscr{N}_{max}(A)$ to $\mathscr{N}_{max}(A/\mathfrak{R}(A))$. Deduce from a) that the intersection of the elements of $\mathscr{N}_{max}(A)$ is equal to $\mathfrak{R}(A)$ and that two arbitrary elements of $\mathscr{N}_{max}(A)$ are transformed into each other by an inner automorphism of A.