Commuting maps and identities with inverses on alternative division rings

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.

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 Torsion Product Property in Alternative Algebras

A celebrated theorem of Wedderburn says that a finite associative division ring is a finite (commutative) field. An elegant proof of this, due to E. Artin [3, p. 72], reduces the problem to showing the reduced (or generic) norm has a nontrivial zero, and then applies the theorem (conjectured by Artin and proved by Chevalley) that a polynomial with coefficients in a finite field and zero constant term has a nontrivial zero in that field if its degree is less than the number of variables. The Wedderburn theorem was generalized by A. A. Albert to finite strictly power-associative division rings of characteristic # 2 in [1, p. 301 ] and [2, p. i1 ] (see also [5 ]). The proof reduced the powerassociative case to the associative case by using the properties of Jordan division algebras. It is the purpose of this note to show that Artin's method carries over directly to the power-associative case using the generic norm introduced by N. Jacobson [4]. By avoiding Jordan algebras one is able to extend Albert's result to the characteristic 2 case. The usual definition of a division ring requires that left and right multiplications by a nonzero element be bijective. If the ring is finite it is necessarily an algebra, over a finite field 4, which is algebraic (even finite-dimensional) and without zero divisors. Also, as we shall see later, the algebra necessarily has a unit element. We wish to consider something slightly more general. For our purposes we say an algebraic power-associative algebra is a division algebra if it is unital and no element xO 0 is a zero divisor in the subalgebra (D[x] it generates (since 4<[x] is finite-dimensional, this is equivalent to saying x is invertible in b [x]). Our conditions are less restrictive, since the Jordan algebra of a quadratic form can be a division algebra in our sense with zero divisors. With this definition we have

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

This paper presents some new rational functional identities on a division ring [Formula: see text] of characteristic not equal to [Formula: see text]. We prove that if the additive maps [Formula: see text] satisfy [Formula: see text], then both [Formula: see text] and [Formula: see text] are of the form [Formula: see text] where [Formula: see text] is an additive map and [Formula: see text] is a fixed element of [Formula: see text]. Further, we show that the only additive maps [Formula: see text] over [Formula: see text] satisfying [Formula: see text] where [Formula: see text] are the zero maps.

CONTENTS Introduction § 0. The analytical and arithmetical hierarchies § 1. Some non-elementary logics § 2. Finite forcing § 3. Rich theories § 4. Some examples of rich theories § 5. Inverse semigroups § 6. Associative rings § 7. Division rings § 8. Open questions References

In this paper a number of theorems are obtained on decompositions of Artinian alternative rings, generalizing analogous theorems of Szász, Šneĭdmjuller, Kertész et al. for associative Artinian rings. For example, it is shown that every alternative Artinian ring decomposes into a direct sum of torsion parts and a torsion free ideal. With several additional restrictions, we obtain a decomposition of any alternative Artinian ring into a direct sum of an ideal and a left ideal, allowing a rather detailed description. A description of hereditarily Artinian alternative rings is given. Bibliography: 16 titles.

There is a particular fascination when two apparently disjoint areas of mathematics turn out to have a meaningful connection to each other. The main goal of this book is to provide a largely self-contained, in-depth account of the linkage between nonassociative algebra and projective planes, with particular emphasis on octonion planes. There are several new results and many, if not most, of the proofs are new. The development should be accessible to most graduate students and should give them introductions to two areas which are often referenced but not often taught. On the geometric side, the book introduces coordinates in projective planes, relates coordinate properties to transitivity properties of certain automorphisms and to configuration conditions. It also classifies higher-dimensional geometries and determines their automorphisms. The exceptional octonion plane is studied in detail in a geometric context that allows nondivision coordinates. An axiomatic version of that context is also provided. Finally, some connections of nonassociative algebra to other geometries, including buildings, are outlined. On the algebraic side, basic properties of alternative algebras are derived, including the classification of alternative division rings. As tools for the study of the geometries, an axiomatic development of dimension, the basics of quadratic forms, a treatment of homogeneous maps and their polarizations, and a study of norm forms on hermitian matrices over composition algebras are included.

Certain functional identities on division rings of characteristic two

Let be a division ring which is not a field with characteristic different from 2 and 3, and with n > 1. The purpose of this paper is to characterize additive maps f,g : satisfying the functional identity f(X) + X 2 g(X −1) = 0 for all invertible .

On maps characterized by action on equal products

Some Pathology for Radicals in Non-Associative Near-Rings

Let R be an associative ring with involution ∗. In this paper we introduce the notion of (α,β)∗-n-derivation in R, where α and β are endomorphisms of R. An additive mapping x↦x∗ of R into itself is called an involution on R if it satisfies the conditions: (i) (x∗)∗=x, (ii) (xy)∗=y∗x∗ for all x,y∈ R. A ring R equipped with an involution ∗ is called a ∗-ring. In the present paper it is shown that if a ∗-prime ring R admits a nonzero (α,β)∗-n-derivation D such that α is surjective, then R is commutative. Some properties of certain n-additive mappings are also discussed in the setting of ∗-prime rings and semiprime ∗-rings. Further, some related properties of (α,β)∗-n-derivation in semiprime ∗-ring have also been investigated. Besides, we have also constructed several examples throughout the text to justify that various restrictions imposed in the hypotheses of our theorems are not superfluous. Finally a structure theorem for (α,β)∗-n-derivation in a semiprime ∗-ring has been established.

We introduce non-associative Ore extensions, S = R[X; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right R δ -linear, where R δ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the nonassociative differential polynomial ring D = R[X; id R , δ]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z( D ) = R δ [p] for a monic p ∈ R δ [X], unique up to addition of elements from Z(R) δ . Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is δ-simple and Z(D) equals the field R δ ∩ Z(R). This provides us with a non-associative generalization of a result by Oinert, Richter and Silvestrov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field R δ ∩ Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.