Skew Poincaré-Birkhoff-Witt Extensions Over Weak Σ-Rigid Rings

Abstract

In this paper we introduce the notion of weak Σ-rigid ring which extends α-rigid rings and Σrigid rings defined for Ore extensions and skew PBW extensions, respectively. We also present the notion of weak Σ-skew Armendariz ring which extends that of α-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.

Some remarks about minimal prime ideals of skew Poincaré-Birkhoff-Witt extensions

“…With the aim of extending the σ-rigid rings defined by Krempa [27], Ouyang [45] introduced the weak σ-rigid rings. For the skew PBW extensions, this weak notion of rigidness was considered by the second author in [54] with the purpose of generalizing the Σ-rigid rings. Let us recall it.…”

“…Therefore, weak Σ-rigid rings are a generalization of Σ-rigid rings to the case where the ring of coefficients is not assumed to be reduced (note that the ring R 3 is not reduced). Nevertheless, from [54], Theorem 3.4, we know that if Σ = {σ 1 , . .…”

“…Section 2 contains the generalization of all the results obtained by Bhat in [5], [6] and [7], from Ore extensions of automorphism type to bijective skew PBW extensions. The Σ-rigid rings and the weak Σ-rigid rings defined by the second author in [47] and [54], respectively, are key in this generalization (they are the corresponding generalization of σ-rigid rings and weak σ-rigid rings, respectively). Also, in this section, we define the Σ( * )-rings as a natural generalization of σ( * )-rings.…”