A Classification of Ring Elements in Skew PBW Extensions Over Compatible Rings

Abstract

For a skew PBW extension over a right duo compatible ring, we characterize several kinds of their elements such as units, idempotent, von Neumann regular, π-regular and the clean elements. As a consequence of our treatment, we extend several results in the literature for Ore extensions and commutative rings.

Skew PBW extensions over symmetric rings

“…Ore extensions of injective type are strictly contained in skew PBW extensions (this is not possible for original PBW extensions; see [32], for a list of noncommutative rings which are skew PBW extensions but not iterated Ore extensions). Several ring and module theoretical properties of skew PBW extensions have been established by different people (e.g., Acosta [1], Artamonov [4], Hamidizadeh et al [15], Hashemi et al [17], [18], [19], Lezama et al [14], [22], [27], [28], [29], [30], [31], [32], [33], Louzari [35], Niño et al [39], [40], Tumwesigye et al [54], Zambrano [55], and the authors [49], [50], [52], [53]). A book containing research results about these extensions has recently been published (see Fajardo et al [11]).…”

“…As a matter of fact, (Σ, ∆)-compatible rings have been very useful in the characterization of different radicals (Wedderburn radical, lower nil radical, Levitzky radical, upper nil radical, the set of all nilpotent elements, the sum of all nil left ideals) and another ring and module theoretical properties of skew PBW extensions (e.g., Hashemi et al [15], [18], [19], and Reyes and Suárez [45], [50]). (…”

$Σ$-semicommutative rings and their skew PBW extensions

“…Regarding the objects of interest in this article, the skew PBW extensions, these objects were defined by Gallego and Lezama [17] with the aim of generalizing families of noncommutative rings as PBW extensions introduced by Bell and Goodearl [8], skew polynomial rings (of injective type) defined by Ore [42], and other as solvable polynomial rings, diffusion algebras, some types of Auslander-Gorenstein rings, some Calabi-Yau and skew Calabi-Yau algebras, some Artin-Schelter regular algebras, some Koszul algebras (see [16] or [46] for a detailed reference to each of these families). Several ring and theoretical properties of skew PBW extensions have been investigated by some people (e.g., Artamonov [5], Fajardo et al [16], Hamidizadeh et al [21], Hashemi et al [22], Lezama [34], and Louzari et al [38]). Precisely, the authors have studied the ring-theoretical notions mentioned above in the setting of skew PBW extensions.…”

Propiedad χ en extensiones PBW torcidas graduadas

“…Gallego y Lezama en [4] definieron una clase especial de anillos de tipo polinomial, los cuales son llamados extensiones PBW torcidas. Varias propiedades de estas extensiones han sido ampliamente estudiadas (véase por ejemplo [5,6,7,9,11,12,13,14,15,16,17,19,21]). Gran parte de los ejemplos, las propiedades y otros aspectos importantes de las extensiones PBW torcidas se encuentran compiladas en [3].…”