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

Abstract

In this paper, we characterize the minimal prime ideals of skew PBW extensions over several classes of rings. We unify different results established in the literature for Ore extensions, and extend all of them to a several families of noncommutative rings of polynomial type which cannot be expressed as these extensions.

On NI and NJ skew PBW extensions

“…Nevertheless, NI and NJ properties have not been studied for skew PBW extensions, so it is a natural task to find necessary or sufficient conditions under which they can be NI and NJ. This is the objective of the paper, and therefore our results contribute to the study of ideals and radicals of skew PBW extensions that has been partially carried out (see [13], [22], [28], [33], [34], and [39]), and establish ring-theoretical properties for noncommutative rings not considered in the literature. As a matter of fact, we generalize some results appearing in Jiang et al, [17], and Nasr-Isfahani [31] and [32].…”

“…. , x n is a skew PBW extension of derivation type over R, then the following assertions hold: [34], Theorem 1, for every completely prime ideal P of A, P ∩ R is a completely prime ideal of R. (iv) If A is NI, then by Proposition 3.2 R is NI, and so N (R) is an ideal of R.…”

Skew PBW extensions over symmetric rings

“…The notion of rigidness with another ring theoretical properties such as minimal prime ideals, Armendariz, McCoy, Baer, quasi-Baer, p. p and p. q have been investigated for skew PBW extensions (c.f. Reyes et al [40], [45], [50]).…”

“…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]).…”

On the differential smoothness of 3-dimensional skew polynomial algebras and diffusion algebras

“…Different physical applications of algebras type 1 and 2 have been studied in the literature. From the point of view of ring-theoretical, homological and computational properties, several papers have been published (see [12], [15], [16], [23], [24], [26], [31], [33], [34], and [37]).…”