Skew Poincar\'e-Birkhoff-Witt extensions over weak $\mathrm{\Sigma }$-rigid rings

In this paper we introduce the notion of weak $\mathrm{\Sigma }$-rigid ring which extends $\alpha$-rigid rings and $\mathrm{\Sigma }$-rigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak $\mathrm{\Sigma }$-skew Armendariz ring which extends that of $\alpha$-skew Armendariz ring. In this way we generalize several results in the literature, from Ore extensions of injective type to the more general setting of skew PBW extensions.