Related papers: On cyclic essential extensions of simple modules o…

Given a set $A$ and an abelian group $B$ with operators in $A$, in the sense of Krull and Noether, we introduce the Ore group extension $B[x; \sigma_B, \delta_B]$ as the additive group $B[x]$, with $A[x]$ as a set of operators. Here, the…

Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E…

We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms…

Given two rings $R \subseteq S$, $S$ is said to be a minimal ring extension of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal extensions of an arbitrary ring $R$, with particular focus on those possessing nonzero…

Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be B{\'e}zout is also studied,…

We calculate the first extension groups for finite-dimensional simple modules over an arbitrary generalized current Lie algebra, which includes the case of loop Lie algebras and their multivariable analogs.

When ${\cal{D}}$ is a linear partial differential operator of any order, a direct problem is to look for an operator ${\cal{D}}_1$ generating the compatibility conditions (CC) ${\cal{D}}_1\eta=0$ of ${\cal{D}}\xi=\eta$. We may thus…

Given a discrete valued field $K$ of positive characteristic, we study the cyclic lifting problem of purely inseparable extensions of the residue field. We prove that unlike the mixed characteristic case, cyclic lifts of any finite purely…

We study base field extensions of ordinary abelian varieties defined over finite fields using the module theoretic description introduced by Deligne. As applications we give algorithms to determine the minimal field of definition of such a…

We prove that a quadratic $A[T]$-module $Q$ with Witt index ($Q/TQ$)$ \geq d$, where $d$ is the dimension of the equicharacteristic regular local ring $A$, is extended from $A$. This improves a theorem of the second named author who showed…

We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local…

We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is…

We define abelian extensions of algebras in congruence-modular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of R-modules for a ring R. We also define a cohomology theory, which we call clone…

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal…

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures…

Our goal is to determine when the trivial extensions of commutative rings by modules are Cohen-Macaulay in the sense of Hamilton and Marley. For this purpose, we provide a generalization of the concept of Cohen-Macaulayness of rings to…

The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie algebra and to prove some elementary properties of $\operatorname{D}$-Lie algebras, the category of $\operatorname{D}$-Lie algebras, the category of modules on a…

We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.

The main aim of this paper is to prove $R$-triviality for simple, simply connected algebraic groups with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field $k$ of arbitrary characteristic. Let $G$ be such a group. We prove…

Let F be a field of characteristic p. We define and investigate nonassociative differential extensions of F and of a central simple division algebra over F and give a criterium for these algebras to be division. As special cases, we obtain…