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Difference between revisions of "Algebra with associative powers"

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A linear algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114101.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114102.png" /> each element of which generates an associative subalgebra. The set of all algebras with associative powers over a given field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114103.png" /> forms a variety of algebras which, if the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114104.png" /> is zero, is defined by the system of identities
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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
 
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$$\label{1}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114105.png" /></td> </tr></table>
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(x,x,x) = (x^2,x,x) = 0
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114106.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114107.png" /> is an infinite field of prime characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114108.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a0114109.png" /> with associative powers of characteristic other than 2 has an idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141011.png" /> can be decomposed according to Peirce into a direct sum of vector subspaces:
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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:
 
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$$\label{2}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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A = A_0(e) \oplus A_{\frac12}(e) \oplus A_1(e)
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141014.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141016.png" /> are subalgebras, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141020.png" />. The decomposition (*) plays a fundamental role in the structure theory of algebras with associative powers.
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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====
 
====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>
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<table>
 
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<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>
 
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<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>
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<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>
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</table>
  
 
====Comments====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011410/a01141021.png" /> was proved in [[#References|[a1]]].
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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 [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Albert,  "On the power associativity of rings"  ''Summa Brasiliensis Math.'' , '''2'''  (1948)  pp. 21–33</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Albert,  "On the power associativity of rings"  ''Summa Brasiliensis Math.'' , '''2'''  (1948)  pp. 21–33</TD></TR>
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</table>
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Revision as of 19:26, 31 October 2016

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

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 [a1].

References

[a1] A.A. Albert, "On the power associativity of rings" Summa Brasiliensis Math. , 2 (1948) pp. 21–33
How to Cite This Entry:
Algebra with associative powers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebra_with_associative_powers&oldid=39571
This article was adapted from an original article by A.T. Gainov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article