  We study two problems involving algebraic properties of Toeplitz operators on
generalized Fock spaces on $\mathbb{C}^d$ with weights of the form
$\left|z\right|^{2s} e^{-\left|z\right|^{2m}}$, $m\geq 1,\ s\geq 0$. We
determine the commutant of a given Toeplitz operator with a radial symbol which
satisfies certain growth conditions. We also discuss the equation $T_fT_g=0$,
when $f$ or $g$ is radial.
