nil and nilpotent ideal

Question

$T_{n} (F)$ is Algebra of upper triangular $n \times n $ matrices over field $ F$.

Can we find an ideal of the algebra $\prod_{n \geq 2} T_{n}(F)$ that is nil but not nilpotent?

Answer 1

Yes: consider, for each $n$, the ideal $I_n$ in $T_n(F)$ of matrices with $0$ on the diagonal. Each element in this ideal has index of nilpotency at most $n$, and this index is attained by matrices with non-zero overdiagonal elements.Thus $I_n$ is a nilpotent ideal, with index of nilpotency exactly $n$.

Now, the ideal: $$I=\bigoplus_{n} I_n\subset \prod_{n\ge 2}T_n(F) $$ is such a non-nilpotent nilideal.

Indeed, any element in this direct sum, being a finite sum of nilpotent elements, is nilpotent. However, it has elements with index of nilpotency as large as we please, so there exists no natural number $N$ such that $$I^N=\bigoplus_{n\ge 2} I_n^N=\bigoplus_{n\ge 2} \{0\}.$$