On types of elements, Gelfand and strongly harmonic rings of skew PBW extensions over weak compatible rings

Abstract We investigate and characterize several kinds of elements such as units, idempotents, von Neumann regular, $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> -regular and clean elements for skew PBW extensions over weak compatible rings. We also study the notions of Gelfand and Harmonic rings for these families of algebras. The results presented here extend those corresponding in the literature for commutative and noncommutative rings of polynomial type.