Ruled minimal surfaces in ${\mathbb{R}}^3$ with density $e^z$

We classify ruled minimal surfaces in ${\mathbb{R}}^3$ with density $e^z\mathrm{.}$ It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in ${\mathbb{R}}^3$ with density $e^z\mathrm{.}$ It is also proved that all translation minimal surfaces are ruled.