Abstract We provide a proof of Jacobson’s theorem on derivations primitive rings with nonzero socle. Both and its formulation (in terms the so-called differential operators left vector spaces over division ring) underlie our paper. apply to describe standard operator real, complex, or quaternionic normed spaces. Indeed, when space is infinite-dimensional, every derivation such ring form $$A\rig...

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were i...

By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type A may be transformed into modules over Lusztig’s infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we f...

Let K be a field of characteristic p > 0, Era the symmetric group on n letters, Sn_1 < Lra the subgroup consisting of the permutations of the first« — 1 letters, and D k the irreducible ATn-module corresponding to a (/^-regular) partition X of n. In [9] we described the socle of the restriction D [T and obtained a number of other results which can be considered as generalisations of the classic...

Let R be a polynomial ring in r variables over an infinite field K, and denote by D a corresponding dual ring, upon which R acts as differential operators. We study type two graded level Artinian algebras A = R/I , having socle degree j. For each such algebra A, we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to ...