Projections and Idempotents in star-reducing Rings Involving the Moore-Penrose Inverse

In [Comput. Math. Appl. 59 (2010) 764-778], Baksalary and Trenkler characterized some complex idempotent matrices of the form $(I-PQ)^{\dag}P(I-Q)$, $(I-QP)^{\dag}(I-Q)$ and $P(P+Q-QP)^{\dag}$ in terms of the column spaces and null spaces of $P$ and $Q$, where $P, Q\in \mathbb{C}^{\textsf{OP}}_n = \{L\in \mathbb{C}_{n,n}\ |\ L^2 = L = L^*\}$. We generalize these results from $\mathbb{C}_{n,n}$ to any *-reducing rings.