In this note we shall investigate a topological version of the problem of Kurosh: “Is any algebraic algebra locally finite?” Kaplansky’s theorem concerning the local nilpotence of nil PI-algebras is well-known. We will prove a generalization of Kaplansky’s theorem to the class of locally compact rings. We use in the proof a theorem of A. I. Shirshov [8] concerning the height of a finitely gener...

The hypersurface in C3 with an isolated quasi-homogeneous elliptic singularity of type Ẽr, r = 6, 7, 8, has a natural Poisson structure. We show that the family of del Pezzo surfaces of the corresponding type Er provides a semiuniversal Poisson deformation of that Poisson structure. We also construct a deformation-quantization of the coordinate ring of such a del Pezzo surface. To this end, we ...

We study the amplitude of mesoscopic Aharonov-Bohm oscillations in quasi-one-dimensional (Q1D) diffusive rings. We consider first the lowtemperature limit of a fully coherent sample. The variance of oscillation harmonics is calculated as a function of the length of the leads attaching the ring to reservoirs. We further analyze the regime of relatively high temperatures, when the dephasing due t...

Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank ≥ 3 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presenta...

In a 1983 paper [M1], I. G. Macdonald introduced his well-known “constant term conjectures.” These conjectures concern a certain polynomial ∆ = ∆(G, k) that is indexed by a semisimple Lie algebra G and a positive integer k. The polynomial ∆ lives in Z[Φ, q], the group ring of the root lattice Φ of G over Z[q]. A basis for this ring, over Z[q], is the set of formal exponentials, ev, for v ∈ Φ th...

We have seen that this map is well-defined and injective; we still have to show that it is surjective. In concrete terms, given a Weil q-integer π, we need to construct a simple abelian variety over Fq on which the Frobenius element acts as π. The only known construction is by reducing a CM abelian variety from characteristic zero. Specifically, our construction will proceed in five steps. Firs...

‎let $r$ be a commutative ring with non-zero identity. ‎we describe all $c_3$‎- ‎and $c_4$-free intersection graph of non-trivial ideals of $r$ as well as $c_n$-free intersection graph when $r$ is a reduced ring. ‎also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎finally, ‎we shall prove that almost all artin rings $r$ have hamiltonian intersection graphs. ...

We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports φ : Z → Spec(R) satisfies the weak Cousin condition if for any integer i the set φ(i) contains all the inmediate generalizations of each point in φ(i+ 1). Every t-structure on Dbfg(R) ...

Abstract Dedekind type DC sums and their generalizations are defined in terms of Euler functions generalization. Recently, Ma et al. (Adv. Differ. Equ. 2021:30 2021) introduced the poly-Dedekind by replacing function appearing sums, they were shown to satisfy a reciprocity relation. In this paper, we consider two kinds new sums. One is unipoly-Dedekind sum associated with 2 unipoly-Euler expres...

If R is a Dedekind domain, P a prime ideal of R and S ⊆R a finite subset then a P -ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490:101–127, 1997), is an ordering {ai}i=1 of the elements of S with the property that, for each 1 < i ≤m, the choice of ai minimizes the P -adic valuation of ∏j<i(s− aj ) over elements s ∈ S. If S, S′ are two finite subsets of R of the same ca...