Converse of the nilpotent zero-product implication

Determine whether, for a -ring and elements aN(R) and bR, the implication that abP(R) entails ab*P(R) implies the converse implication that ab=0 entails ab*=0.

Background

For a nil--reversible ring, the paper studies two related conditions for a nilpotent element a and an arbitrary element b: the zero-product condition ab=0 ab*=0 and the projection condition abP(R) ab*P(R). The authors prove the implication from the zero-product condition to the projection condition in Proposition 4.12. They explicitly leave unresolved whether the projection condition is strong enough to recover the zero-product condition, which would establish the converse implication.

References

We now give a proof for the following implication:

ab=0\text{ implies }ab{\ast}=0 \quad\Longrightarrow\quad ab\in P(R)\text{ implies }ab{\ast}\in P(R).

At present, it remains unknown whether the converse implication holds.

— On $\ast$-Reversible and Generalized $\ast$-Reversible Rings  (2609.20076 - Chen et al., 17 Sep 2026) in Section 4, immediately before Proposition 4.12