Algebra with associative powers

A linear algebra $A$ over a field $F$ each element of which generates an associative subalgebra. The set of all algebras with associative powers over a given field $F$ forms a variety of algebras which, if the characteristic of the field $F$ is zero, is defined by the system of identities $$\label{1} (x,x,x) = (x^2,x,x) = 0$$ where the associator $(a,b,c) = (ab)c - a(bc)$. If $F$ is an infinite field of prime characteristic $p$, then the variety of algebras with associative powers cannot be defined by any finite system of identities, but an independent, infinite system of identities which defines it is known [3]. If a commutative algebra $A$ with associative powers of characteristic other than $2$ has an idempotent $e \neq 0$, then $A$ can be decomposed according to Peirce into a direct sum of vector subspaces: $$\label{2} A = A_0(e) \oplus A_{\frac12}(e) \oplus A_1(e)$$ where $A_\lambda(e) = \{ a \in A : ea = \lambda a \}$, $\lambda = 0,\frac12,1$. Here $A_0(e)$ and $A_1(e)$ are subalgebras, $A_0(e) A_1(e) = 0$, $A_{\frac12}(e)A_{\frac12}(e) \subseteq A_0(e) + A_1(e)$, $A_\lambda(e) A_{\frac12}(e) \subseteq A_{\frac12}(e) + A_{1-\lambda}(e)$ for $\lambda = 0,1$. The Pierce decomposition (2) plays a fundamental role in the structure theory of algebras with associative powers.

References

 [1] A.A. Albert, "Power-associative rings" Trans. Amer. Math. Soc. , 64 (1948) pp. 552–593 [2] A.T. Gainov, "Identity relations for binary Lie rings" Uspekhi Mat. Nauk , 12 : 3 (1957) pp. 141–146 (In Russian) [3] A.T. Gainov, "Power-associative algebras over a finite-characteristic field" Algebra and Logic , 9 : 1 (1970) pp. 5–19 Algebra i Logika , 9 : 1 (1970) pp. 9–33

An algebra with associative powers is also called a power-associative algebra. The fact that the set of algebras with associative powers over a field of non-zero characteristic forms a variety defined by (1) $(x,x,x) = (x^2,x,x) = 0$ was proved in [a1].