Associated prime ideals over skew PBW 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].…”

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

“…Last, but not least, we consider as a possible future work to investigate minimal prime ideals and the 2-primal property in a more general context of noncommutative rings than skew PBW extensions such as for example the semi-graded rings introduced by Lezama [33] (see also [29] and [32]), or maybe considering a weak notion of compatibility following the ideas presented recently by the second author [57], and of course, in the setting of modules over these extensions, see [36], [42], and [48].…”

“…Some words about the generality of skew PBW extensions with respect to another families of noncommutative rings are said in Section 1. Homological and ring-theoretical properties for these extensions have been investigated by several people (e.g., Artamonov [2], Hashemi et al, [17], [18], [19], and [20], Lezama et al, [1], [24], [29], [31], [33], [35], Louzari et al, [37], Tumwesigye et al, [61], Zambrano [62], and the authors, [42], [43], [48], [51], [52] and [55]). As a matter of fact, a book containing research results about these extensions has recently been published, see [10].…”

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

“…They obtained different properties of modules over the ring of coefficients and the corresponding Ore extension. Now, recently, the notion of (Σ, ∆)-compatibility in the context of modules over skew PBW extensions has been considered by the third author in [51] with the aim of obtaining similar results to those established in [1] for Ore extensions, and also in [46] with the purpose of characterizing the associated prime ideals over these extensions generalizing the treatment developed in [3] for Ore extensions. Having this in mind and considering the results obtained in this paper, we think as a future work to investigate a classification of several types of elements in modules over these extensions.…”