Exercise 6, § 10

Let A be a Artinian ring and $n=$ long$_A(A_s)$.

a) Prove that the ring A is $n$-regular (Exercise 5).

b) Prove that the ring A is pseudoregular. (Let $r$ be an integer such that $\Re (A)^r= 0$, and let $m$ be the length of the ring $B = A/\Re (A)$. Prove, by reducing to the case when the ring B is simple, that for every $b\in B$, there exists an element $y$ of B that commutes with $b$ and satisfies $b^{m+1}y=b^m$. Deduce that for every $a\in A$, there exists an $x\in A$ such that $a^{mr}xa^{mr}=a^{mr}.$)

c) Let $\mathfrak{I}$ be the set of nonnilpotent left ideals of A. Prove that the minimal elements of $\mathfrak{I}$ are the ideals $Ae$ generated by an indecomposable idempotent (use a) and Exercise 5, c)). Prove that every left ideal of A is the direct sum of finitely many ideals of this type and a left ideal contained in the radical.

d) A left ideal $\mathfrak{a}$ of A is generated by an idempotent if and only if it is the direct sum of a finite family of left ideals $(Ae_i)_{i\in I}$, where the $e_i$ are indecomposable orthogonal idempotents.