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The polynomial rings over hereditary Noetherian prime rings have global dimension 2 and any two-sided ideal which is either left -ideal or right -ideal is left and right projective. By using the latter property, we define the concept of generalized hereditary Noetherian prime rings (G-HNP rings). We study the structure of projective ideals in G-HNP rings and some overrings of G-HNP rings. As it is shown in the examples, the class of G-HNP rings ranges from rings with global dimension 2 to rings with infinite global dimension and Noetherian prime rings with global dimension 2 are not necessarily G-HNP rings. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic zero, and let Γ be an additive subgroup of k . Results of Kaplansky–Santharoubane and Su classify intermediate series representations of the generalised Witt algebra W Γ in terms of three families, one parameterised by A 2 and two by P 1 . In this note, we use the first family to construct a homomorphism Φ from the enveloping algebra U ( W Γ ) to a skew extension k [ A 2 ] ⋊ Γ of the coordinate ring of A 2 . We show that the image of Φ is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of U ( W Γ ) under Φ is not left or right noetherian, giving a new proof that U ( W Γ ) is not noetherian. We construct Φ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let G be an arbitrary group and let A be a G -graded ring. A graded A -module M is an intermediate series module if M g is one-dimensional for all g ∈ G . Given a shift-invariant family of intermediate series A -modules parametrised by a scheme X , we construct a homomorphism Φ from A to a skew extension of k [ X ] . The kernel of Φ consists of those elements which annihilate all modules in X . [ABSTRACT FROM AUTHOR]

For a Kurosh–Amitsur radical class of rings, we investigate the existence, for a radical subringSof a ringA, of a largest subringTofAfor whichSis the radical. WhenTexists, it is called the radicalizer ofS. There are no radical classes of associative rings for which every radical subring of every ring has a radicalizer. If a subring is the radical of its idealizer, then the idealizer is a radicalizer. We examine radical classes for which each radical subring is contained in one which is the radical of its own idealizer. [ABSTRACT FROM PUBLISHER]