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In this paper we study skew Poincar\'e-Birkhoff-Witt extensions over weaksymmetric and $(\Sigma,\Delta)$-weak symmetry rings. Since these extensionsgeneralize Ore extensions of injective type and another noncommutative rings ofpolynomial type, we unify and extend several results in the literatureconcerning the property of being symmetry. Under adequate conditions, wetransfer the property of being weak symmetric or $(\Sigma,\Delta)$-weaksymmetric from a ring of coefficients to a skew PBW extension over this ring.We illustrate our results with remarkable examples of algebras appearing innoncommutative algebraic geometry and theoretical physics.

Skew PBW extensions over weak symmetric and $(\Sigma,\Delta)$-weak symmetric rings