A note on products of idempotents in the ring of upper triangular infinite matrices (Q6649808)

Every \(n\times n\) matrix \(A\) over a field \(F\) can be written as a product of \(n\) idempotent matrices, but in general no fewer (see, for example, [\textit{C. S. Ballantine}, Linear Algebra Appl. 19, 81--86 (1978; Zbl 0371.15002)]). Let \(\mathcal{T}_{\infty }\left( F\right) \) be the ring of all \(\mathbb{N\times N}\) matrices over \(F\). The author of the present paper considers the question of whether \(A\in \mathcal{T}_{\infty }\left( F\right) \) can be written as a product of finitely many idempotents in this ring and provides general examples where \(A\) is not such a product. On the other hand, if \(A\) is a product of finitely many nilpotent matrices in \(\mathcal{T} _{\infty }\left( F\right) \), then \(A\) is also a product \(T_{1} \cdots T_{k}\) of idempotent matrices \(T_{i}\) in which each column has only finitely many nonzero entries.