A new characterization of GCD domains of formal power series

By using the $v$-operation, a new characterization for a power series ring to be a GCD domain is discussed. It is shown that if $D$ is a $\mathrm{UFD}$, then $D[\![X]\!]$ is a GCD domain if and only if for any two integral $v$-invertible $v$‑ideals $I$ and $J$ of $D[\![X]\!]$ such that $(IJ)_{0}\neq (0),$ we have $((IJ)_{0})_{v}$ $= ((IJ)_{v})_{0},$ where $I_0=\{f(0) \mid f\in I\}$. This shows that if $D$ is a GCD domain such that $D[\![X]\!]$ is a $\pi$-domain, then $D[\![X]\!]$ is a GCD domain.