On Jordan ideals and matrix rings

For A, a commutative ring, and results by Costa and Keller characterize certain -normalized subgroups of the symplectic group, via structures utilizing Jordan ideals and the notion of radices. The following work creates a Jordan ideal structure theorem for -graded rings, A₀A₁, and a -graded matrix algebra. The major theorem is a generalization of Costa and Keller’s previous work on matrix algebras over commutative rings.     

Ternary Derivations of Generalized Cayley–Dickson Algebras

Abstract

Ternary derivations, ternary Cayley derivations and ternary automorphisms are computed over fields of characteristic ≠ 2, 3 for the algebras A _t obtained by the Cayley–Dickson duplication process. While the derivation algebra of A _t stops growing after t = 3, the ternary derivation algebra significantly decreases in the step from the octonions A ₃ to the sedenions A ₄, revealing the symmetry lost on that stage.     

Sur la Classification des Nilalgèbres Commutatives de Nilindice 3

We give some structure results and recursive-like methods for constructions and classifications of commutative nilalgebras of nilindex 3.     

Extending Rings of Prüfer Type in Central Simple Algebras

Let R be a commutative integral domain with field of fractions F and let Q be a finite-dimensional central simple F-algebra. If R is a Prüfer domain then it is still unknown whether or not R can be extended to a Prüfer order in Q in the sense of Alajbegovi? and Dubrovin (J. Algebra, 135: 165–176, 1990). In this paper we investigate a more general class of rings which we call rings of Prüfer type and we will prove an extension theorem for these rings. Under special assumptions this result also leads to an extension theorem for certain Prüfer domains.     

Noncoherence of some rings of functions

Let , denote the unit disc and unit circle, respectively, in , with center 0. If , then let denote the set of complex-valued functions defined on that are analytic in , and continuous and bounded on . Then is a ring with pointwise addition and multiplication. We prove that if the intersection of with the set of limit points of is not empty, then the ring is not coherent.

     

Hyperbolic unit groups and quaternion algebras

We classify the quadratic extensions and the finite groups G for which the group ring [G] of G over the ring of integers of K has the property that the group of units of augmentation 1 is hyperbolic. We also construct units in the ℤ-order of the quaternion algebra , when it is a division algebra.     

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Let K be an ultrametric complete field and let E be an ultrametric space. Let A be the Banach K-algebra of bounded continuous functions from E to K and let B be the Banach K-algebra of bounded uniformly continuous functions from E to K. Maximal ideals and continuous multiplicative semi-norms on A (resp. on B) are studied by defining relations of stickiness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of A (resp. of B) is in bijection with the set of equivalence classes with respect to stickiness (resp. to contiguousness). Every prime ideal of A or B is included in a unique maximal ideal and every prime closed ideal of A (resp. of B) is a maximal ideal, hence every continuous multiplicative semi-norms on A (resp. on B) has a kernel that is a maximal ideal. If K is locally compact, every maximal ideal of A (resp. of B) is of codimension 1. Every maximal ideal of A or B is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, on A as on B the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of A, Max(A) and the Banaschewski compactification of E which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of A (resp. B) is equal to the whole set of continuous multiplicative semi-norms.     

Algebras, hyperalgebras, nonassociative bialgebras and loops

Sabinin algebras are a broad generalization of Lie algebras that include Lie, Malcev and Bol algebras as very particular examples. We present a construction of a universal enveloping algebra for Sabinin algebras, and the corresponding Poincaré-Birkhoff-Witt Theorem. A nonassociative counterpart of Hopf algebras is also introduced and a version of the Milnor-Moore Theorem is proved. Loop algebras and universal enveloping algebras of Sabinin algebras are natural examples of these nonassociative Hopf algebras. Identities of loops move to identities of nonassociative Hopf algebras by a linearizing process. In this way, nonassociative algebras and Hopf algebras interlace smoothly.     

Construction of dialgebras through bimodules over algebras

The varieties of dialgebras (also known as Loday-type algebras) over a given type of algebra have been the subject of multiple recent developments. We provide here a construction of such dialgebra varieties via bimodules over an algebra and a surjective equivariant map. Our construction is equivalent to the KP construction (Kolesnikov–Pozhidaev construction) when departing from the set of linearized identities of the algebra variety. The novel construction simplifies the obtention of the dialgebra equations without forcing a complete linearization of the algebra identities. We illustrate the use of the novel construction providing the dialgebras associated to several varieties of algebras, including those over diverse Lie admissible algebras. We provide some novel explorations on the structure of the dialgebras which are easily articulated through our construction.     

Trees and valuation rings

A subring of a division algebra is called a valuation ring of if or holds for all nonzero in . The set of all valuation rings of is a partially ordered set with respect to inclusion, having as its maximal element. As a graph is a rooted tree (called the valuation tree of ), and in contrast to the commutative case, may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra , and one main result here is a positive answer to this question where can be chosen as a quaternion division algebra over a commutative field.

     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

Sum of Element Orders of Maximal Subgroups of the Symmetric Group

For a finite group G, let ψ(G) denote the sum of element orders of G. The aim of this article is to show that ψ(H) < ψ(A _n ) for every proper subgroup H of the symmetric group of degree n, which is different from the alternating group A _n .