Alternative Division Rings Research Articles

Commuting maps and identities with inverses on alternative division rings

Let D be an alternative division ring with characteristic different from two. The purpose of this paper is to characterize additive mappings f,g:D→D satisfying certain identities studied by Vukman, Brešar, and Catalano previously on an associative division ring. We also present some new identities concerning Lie and Jordan products, and provide a complete description of commuting maps on octonion algebras.

Variations on a Theme of Glauberman

A new and elementary proof of the Artin–Zorn theorem that finite alternative division rings are fields is given. The characterisation of finite fields of Glauberman and Heimbeck is also extended to a broader class of fields, the two subjects being connected via geometry.

Automorphisms on the alternative division ring

In this work, we prove that, if $\mathfrak{R} $ is an arbitrary alternative division ring, then, under a mild condition, every semi-automorphism $\varphi : \mathfrak{R} \rightarrow \mathfrak{R} $ is either an automorphism or an anti-automorphism. We extend Hua's result L.K. Hua for an alternative division ring.

Projective embeddings of dual polar spaces of mixed type

The classification of polar spaces was completed by Tits in his monumental work [19] who showed as an intermediate step that almost all of them were embeddable in a projective space. The current paper is part of a project to classify all full projective embeddings of the duals of polar spaces. A complete classification is available for rank 2 by the work of Tits [19], Buekenhout–Lefèvre [4] and Dienst [13]. Recently, significant process was made in the rank 3 case by De Bruyn and Van Maldeghem [12], who showed, among other things, that the members of five families of dual polar spaces of rank 3 related to alternative division rings have full projective embeddings. For two of these families, the quaternionic dual polar spaces and the dual polar spaces of mixed type, it was not yet known whether full projective embeddings exist for rank at least four. In the present paper, we prove that any “mixed dual polar space” of rank at least 2 has a full projective embedding.

Projective embeddings of dual polar spaces arising from a class of alternative division rings

Abstract We discuss some recent results of us regarding a class of polar spaces which includes the nonembeddable polar spaces introduced by Tits [Tits, J., “Buildings of spherical type and finite BN-pairs,” Lecture Notes in Mathematics 386 , Springer-Verlag, Berlin-New York, 1974]. These results include an elementary construction of the polar space, a construction of a polarized embedding of the corresponding dual polar space and the determination whether this projective embedding is universal and unique (as a polarized embedding).

ON THE CLASSIFICATION OF CENTRALLY FINITE ALTERNATIVE DIVISION RINGS SATISFYING ALGEBRAIC CLOSURE CONDITIONS

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

A Generalization of Rings of Type (1, 1)

x , y , z s x , z , y 1 Ž . Ž . Ž . Ž . w x Ž . have been studied. Among these are rings of type 1, 1 2 , which satisfy 1 Ž . Ž . w x and x, x, x s 0, third power associativity , Novikov rings 4, 6, 7, 10 , Ž . Ž . Ž . which satisfy 1 and x yz s y xz , and more recently rings which satisfy Ž . Ž w x . w x 1 and w, y, z , s s 0, 1, 8 . In the present paper we study rings Ž . Ž Ž . . Ž . satisfying 1 and w, x, x, x , y s 0. This generalizes rings of type 1, 1 . Ž w x . w 2 x Ž . It also generalizes the identity w, y, z , s s 0, since x , x s x, x, x . The result we obtain is that simple rings of characteristic / 2, 3, 5 must be associative. There exist examples of simple Novikov rings which are not w x associative 3 , and there exist examples of right alternative division rings Ž Ž .. w x of characteristic 2 thus of type 1, 1 which are not associative 9 .

The Torsion Product Property in Alternative Algebras

In an alternative ring, when is the product of torsion units a torsion unit? We answer this question completely for alternative division rings, vector matrix algebras, and loop algebras.

On locally finite alternative division rings with valuation

We prove that a locally finite complete alternative division ring with valuation is a field, without using the celebrated theorem of Bruck and Kleinfeld, which classifies all division rings with IP.

Compact 16-dimensional projective planes with large collineation groups. II

If a locally compact topological projective planeP of finite topological dimension satisfies dim AutP > 46, thenP is isomorphic to the Moufang plane over the alternative division ring $$\mathbb{O}$$ of the real octonions.

The isostrophisms of planar ternary rings part I

It is well known that a projective plane nay be coordinatized by picking four arbitrary, ordered points such that no three of them are collinear. This gives rise to an algebraic structure called the planar ternary ring T with respect to the base points A,B,C,D. The existence of certain collineations in the plane will be reflected by algebraic properties in T, and conversely. In this paper the interrelationships between certain algebraic properties in T, and the linearity of those ternary rings which are obtained from T and various permutations of A,B,C,D are considered. Many well known types of algebraic structures, such as quasifields and alternative division rings will occur if the ternary rings generated by suitable permutations of A,B,C,D are linear.

Direct Product Decomposition of Alternative Rings

It is shown that any alternative ring $A$ equipped with the relation $\leqslant$, defined by $x \leqslant y$ if and only if $xy = {x^2}$, is isomorphic to a direct product of alternative division rings if and only if the relation $\leqslant$ is a partial order on $A$ such that $A$ is hyperatomic and orthogonally complete.

Direct product decomposition of alternative rings

It is shown that any alternative ring A A equipped with the relation ⩽ \leqslant , defined by x ⩽ y x \leqslant y if and only if x y = x 2 xy = {x^2} , is isomorphic to a direct product of alternative division rings if and only if the relation ⩽ \leqslant is a partial order on A A such that A A is hyperatomic and orthogonally complete.

Right Bol Quasi-Fields

We shall consider quasi-fields which satisfy the multiplicative Identity1.1(1.1) will be called the right Bol law and a quasi-field satisfying it will be called a right Bol quasi-held. Moufang quasi-fields, i.e., those satisfying the Moufang identity1.2were studied in (5). Quasi-fields satisfying the left Bol identity1.3were studied by Burn (3) and the author (6). Such quasi-fields are called Bol quasi-fields.Our investigation will parallel the investigations in (5; 6). In § 2 we derive necessary and sufficient conditions for a right Bol quasi-field to be an alternative division ring and also criteria for it to be a near-field. With this information we derive in §§ 3 and 4 new characterizations of Moufang planes similar to those in (5; 6).Loops satisfying (1.1) have been studied by Robinson (10). He calls such loops Bol loops.

On the left nucleus of a Bruck ring

is one-to-one of F upon F. (See [2],' Appendix.) R. H. Bruck has shown that fields F having additive endomorphisms 0 satisfying (1.2), and yet not right multiplications, actually exist. The author, in [I], has therefore called a not alternative, right alternative division ring B (necessarily of characteristic two) a Bruck ring if it is two dimensional over some field F and if multiplication in B is given by (1.1). The field F turns out to be the left nucleus of B and, in all the examples given by Bruck, is a simple transcendental extension of a field of characteristic two, and thus imperfect. In this note we give an elementary proof that this situation must of necessity occur:

Right Alternative Division Rings of Characteristic Two

Right Alternative Division Rings of Characteristic Two

Right alternative division rings of characteristic two

Right alternative division rings of characteristic two

An Extension of the Theorem on Alternative Division Rings

An Extension of the Theorem on Alternative Division Rings

An extension of the theorem on alternative division rings

An extension of the theorem on alternative division rings

Alternative Division Rings of Characteristic 2

Alternative Division Rings of Characteristic 2