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Difference between revisions of "Algebraic algebra"

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An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114201.png" /> with associative powers (in particular, an associative algebra) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114202.png" /> all elements of which are algebraic (an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114203.png" /> is called algebraic if the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114204.png" /> generated by it is finite-dimensional or, equivalently, if the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114205.png" /> has an annihilating polynomial with coefficients from the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114206.png" />). An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114207.png" /> is called an algebraic algebra of bounded degree if it is algebraic and if the set of degrees of the minimal annihilating polynomials of its elements is bounded. Subalgebras and homomorphic images of an algebraic algebra (of bounded degree) are algebraic algebras (of bounded degree).
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An [[algebra with associative powers]] (in particular, an [[associative algebra]]) over a field in which all elements are algebraic: an element $a$ of the algebra $A$ is called ''algebraic'' over the field $F$ if the subalgebra $F[a]$ generated by $a$ is finite-dimensional or, equivalently, if the element $a$ has an annihilating polynomial with coefficients from the ground field $F$. An algebra $A$ is called an ''algebraic algebra of bounded degree'' if it is algebraic and if the set of degrees of the minimal annihilating polynomials of its elements is bounded. Subalgebras and homomorphic images of an algebraic algebra (of bounded degree) are algebraic algebras (of bounded degree).
  
Examples: locally finite algebras (in particular, finite-dimensional ones), nil-algebras and associative skew-fields with a countable set of generators over an uncountable field.
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Examples: [[locally finite algebra]]s (in particular, finite-dimensional ones), [[nil algebra]]s and associative [[skew-field]]s with a countable set of generators over an uncountable field.
  
The algebras considered below are associative. The [[Jacobson radical|Jacobson radical]] of an algebraic algebra is a nil-ideal. A primitive algebraic algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114208.png" /> is isomorphic to a dense algebra of linear transformations of a vector space over a skew-field; if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a0114209.png" /> is of bounded degree, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011420/a01142010.png" /> is isomorphic to a ring of matrices over a skew-field. An algebraic algebra without non-zero nilpotent elements (in particular, a skew-field) over a finite field is commutative. It follows that finite skew-fields are commutative. An algebraic algebra of bounded degree satisfies a polynomial identity (cf. [[PI-algebra|PI-algebra]]). An algebraic PI-algebra is locally finite. If the ground field is uncountable, then the algebras obtained from an algebraic algebra by extension of the ground field, and the tensor product of algebraic algebras, are algebraic algebras.
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The algebras considered below are associative. The [[Jacobson radical]] of an algebraic algebra is a [[nil ideal]]. A primitive algebraic algebra $A$ is isomorphic to a dense algebra of linear transformations of a vector space over a skew-field; if, in addition, $A$ is of bounded degree, then $A$ is isomorphic to a ring of matrices over a skew-field. An algebraic algebra without non-zero nilpotent elements (in particular, a skew-field) over a [[finite field]] is commutative. It follows that finite skew-fields are commutative. An algebraic algebra of bounded degree satisfies a polynomial identity (cf. [[PI-algebra]]). An algebraic PI-algebra is locally finite. If the ground field is uncountable, then the algebras obtained from an algebraic algebra by extension of the ground field, and the tensor product of algebraic algebras, are algebraic algebras.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.N. Herstein,  "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  I.N. Herstein,  "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR>
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</table>

Latest revision as of 16:55, 16 October 2014

An algebra with associative powers (in particular, an associative algebra) over a field in which all elements are algebraic: an element $a$ of the algebra $A$ is called algebraic over the field $F$ if the subalgebra $F[a]$ generated by $a$ is finite-dimensional or, equivalently, if the element $a$ has an annihilating polynomial with coefficients from the ground field $F$. An algebra $A$ is called an algebraic algebra of bounded degree if it is algebraic and if the set of degrees of the minimal annihilating polynomials of its elements is bounded. Subalgebras and homomorphic images of an algebraic algebra (of bounded degree) are algebraic algebras (of bounded degree).

Examples: locally finite algebras (in particular, finite-dimensional ones), nil algebras and associative skew-fields with a countable set of generators over an uncountable field.

The algebras considered below are associative. The Jacobson radical of an algebraic algebra is a nil ideal. A primitive algebraic algebra $A$ is isomorphic to a dense algebra of linear transformations of a vector space over a skew-field; if, in addition, $A$ is of bounded degree, then $A$ is isomorphic to a ring of matrices over a skew-field. An algebraic algebra without non-zero nilpotent elements (in particular, a skew-field) over a finite field is commutative. It follows that finite skew-fields are commutative. An algebraic algebra of bounded degree satisfies a polynomial identity (cf. PI-algebra). An algebraic PI-algebra is locally finite. If the ground field is uncountable, then the algebras obtained from an algebraic algebra by extension of the ground field, and the tensor product of algebraic algebras, are algebraic algebras.

References

[1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2] I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)
How to Cite This Entry:
Algebraic algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_algebra&oldid=15135
This article was adapted from an original article by V.N. Latyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article