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In this note we show that $B$-scrolls over null curves in a 3-dimensional Lorentzian space form ${\overline{M}}_1^3\left(c\right)$ are characterized as the only ruled surfaces with null rulings whose Gauss maps $G$ satisfy the condition $\Delta G=\Lambda G$, $\Lambda X\left(\overline{M}\right)\to X\left(\overline{M}\right)$ being a parallel endomorphism of $X\left(\overline{M}\right)$.