View source for 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 [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Albert,   "Power-associative rings"  ''Trans. Amer. Math. Soc.'' , '''64'''  (1948)  pp. 552–593</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.T. Gainov,   "Identity relations for binary Lie rings"  ''Uspekhi Mat. Nauk'' , '''12''' :  3  (1957)  pp. 141–146  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR>
</table>

====Comments====
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 {{Cite|a1}}.

====References====

* {{Ref|a1}} A.A. Albert,   "On the power associativity of rings"  ''Summa Brasiliensis Math.'' , '''2'''  (1948)  pp. 21–33 {{ZBL|0039.26403}}

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