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 $(I-PQ{)}^{\mathrm{†}}P(I-Q)$, $(I-QP{)}^{\mathrm{†}}(I-Q)$ and $P(P+Q-QP{)}^{\mathrm{†}}$ in terms of the column spaces and null spaces of $P$ and $Q$, where $P, Q\in $\mathbb{C}$^{$\mathsf{\text{OP}}$}n = {L\in $\mathbb{C}${n,n}\ |\ L^2 = L = L^*}$. We generalize these results from ${\mathbb{C}}_{n,n}$ to any *-reducing rings.