Half-axes in power associative algebras

ON m-STABLE COMMUTATIVE POWER- ASSOCIATIVE ALGEBRAS

A commutative power-associative algebra A of characteristic >5 with an idempotent u may be written1 as the supplementary sum ^=^4„(l)+4u(l/2)+^4u(0) where 4U(X) is the set of all xx in A with the property xx« =Xxx. The subspaces Au(l) and .4K(0) are orthogonal subalgebras, [AU(1/2)]2QAU(1)+AU(0) andAu(K)Au(l/2) C4„(l/2)+^4u(l—X) forX=0, 1. The algebra A is called w-stable if 4u(X)-4„(l/2)C.4u(l...

Antiflexible Algebras Which Are Not Power-associative

Power-associative antiflexible algebras have been studied by the author [3] and Kosier [2]. As a matter of terminology, we shall define an algebra as a finite dimensional vector space on which a multiplication is defined in which both distributive laws are satisfied. If A is an algebra over a field F of characteristic not two, then A has an attached algebra A + which is the same additive group ...

Hyperidentities in Associative Graph Algebras

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G is called associative if the corresponding graph algebra A(G) satisfies the equation (xy)z ≈ x(yz). An identity s ≈ t of terms s and t of any type τ ...

Identities of Associative Algebras

The structure theory for Pi-algebras is well developed. Some results of this theory are classic now. One of them is Kaplansky's theorem which asserts that a primitive Pi-algebra is finite dimensional over its centre. Another example is the theorem of Nagata-Higman which asserts that any algebra over a field of zero characteristic satisfying identity x" = 0 is nilpotent. In 1957 A.I. Shirshov pr...

Non Associative Linear Algebras