  In this note, we study the generalized fraction properties and power series
properties of $\mathcal{S}$-Noetherian rings. Actually, we investigate two
questions proposed in [A. Dabbabi, A. Benhissi, Generalization of the
$S$-Noetherian concept, {\it Arch. Math.} (Brno) \textbf{59}(4) (2023)
307-314.]
