Abstract
In this paper, we characterize the units of skew PBW extensions over compatible rings. With this aim, we recall the transfer of the property of being 2-primal for these extensions. As a consequence of our treatment, the results established here generalize those corresponding for commutative rings and Ore extensions of injective type. In this way, we present new results for several noncommutative rings of polynomial type not considered before in the literature.
Highlights
Throughout the paper, for a ring B, the lower nil radical or the prime radical, the upper radical, the set of nilpotent elements of B, and the Jacobson radical of B are denoted by Nil∗(B), Nil∗(B), Nil(B) and J(B), respectively
The importance of 2-primal rings is that they can be considered as a generalization of commutative rings and reduced rings
Considering all above results and with the aim of extending of all them to the more general setting of skew PBW extensions introduced by Gallego and Lezama [9], Hashemi et al, [12] studied under certain conditions the connections of the prime radical and the upper nil radical of a ring R with the prime radical and the upper nil radical of a skew PBW extension A over R
Summary
Throughout the paper, for a ring B, the lower nil radical or the prime radical (the intersection of all prime ideals in B), the upper radical (the sum of all nil ideals), the set of nilpotent elements of B, and the Jacobson radical of B are denoted by Nil∗(B), Nil∗(B), Nil(B) and J(B), respectively. Considering all above results and with the aim of extending of all them to the more general setting of skew PBW extensions introduced by Gallego and Lezama [9] (in Section 2, we say a few words about these objects), Hashemi et al, [12] studied under certain conditions the connections of the prime radical and the upper nil radical of a ring R with the prime radical and the upper nil radical of a skew PBW extension A over R They considered the transfer of several properties such as being prime, semiprime and the characterization of minimal prime ideals. Our paper can be considered as a modest contribution to [12] and to the study of the 2-primal property and minimal prime ideals for noncommutative rings of polynomial type which can not be expressed as Ore extensions of injective type. We recall some results about skew PBW extensions and (Σ, ∆)-compatible rings which are important for the rest of the paper
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