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Difference between revisions of "Free associative algebra"

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The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415201.png" /> of polynomials over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415202.png" /> in non-commuting variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415203.png" />. The following universal property determines the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415204.png" /> uniquely up to an isomorphism: There is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415205.png" /> such that any mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415206.png" /> into an associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415207.png" /> with a unit over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415208.png" /> can be factored through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f0415209.png" /> in a unique way. The basic properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152010.png" /> are:
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The algebra $k\langle X \rangle$ of polynomials over a field $k$ in non-commuting variables in $X$. The following universal property determines the algebra $k\langle X \rangle$ uniquely up to an isomorphism: There is a mapping $i : k \rightarrow k\langle X \rangle$ such that any mapping from $X$ into an associative algebra $A$ with a unit over $k$ can be factored through $k\langle X \rangle$ in a unique way. The basic properties of $k\langle X \rangle$ are:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152011.png" /> can be imbedded in a skew-field (the Mal'tsev–Neumann theorem);
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1) $k\langle X \rangle$ can be imbedded in a skew-field (the Mal'tsev–Neumann theorem);
 
 
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152012.png" /> has a weak division algorithm, that is, the relation
 
 
 
<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/f/f041/f041520/f04152013.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152014.png" />, all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152015.png" /> are non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152017.png" />, always implies that there are an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152019.png" />, and elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152020.png" /> such that
 
 
 
<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/f/f041/f041520/f04152021.png" /></td> </tr></table>
 
  
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2) $k\langle X \rangle$ has a weak division algorithm, that is, the relation
 +
$$
 +
d \left({ \sum_{i=1}^n a_i b_i }\right) < \max_i \{ d(a_i) + d(b_i) \}
 +
$$
 +
where $a_i, b_i \in k\langle X \rangle$, all the $a_i$ are non-zero ($i = 1,\ldots,n$), $d(a_1) \le \cdots \le d(a_n)$, always implies that there are an integer $r$, $1 < r \le n$, and elements $c_,\ldots,c_{r-1}$ such that
 +
$$
 +
d\left({ a_r - \sum_{i=1}^{r-1} a_i c_i }\right) < d(a_r)
 +
$$
 
and
 
and
 +
$$
 +
d(a_i) + d(c_i) < d(a_r),\ \ \ i=1,\ldots,r-1
 +
$$
 +
(here $d(a)$ is the usual degree of a polynomial $a \in k\langle X \rangle$, $d(0) = -\infty$);
  
<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/f/f041/f041520/f04152022.png" /></td> </tr></table>
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3) $k\langle X \rangle$ is a left (respectively, right) free ideal ring (that is, any left (respectively, right) ideal of $k\langle X \rangle$ is a free module of uniquely determined rank);
  
(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152023.png" /> is the usual degree of a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152025.png" />);
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4) the centralizer of any non-scalar element of $k\langle X \rangle$ (that is, the set of elements that commute with a given element) is isomorphic to the algebra of polynomials over $k$ in a single variable (Bergman's theorem).
 
 
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152026.png" /> is a left (respectively, right) free ideal ring (that is, any left (respectively, right) ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152027.png" /> is a free module of uniquely determined rank);
 
 
 
4) the centralizer of any non-scalar element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152028.png" /> (that is, the set of elements that commute with a given element) is isomorphic to the algebra of polynomials over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041520/f04152029.png" /> in a single variable (Bergman's theorem).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1971)</TD></TR></table>
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<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR>
'
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1971)</TD></TR>
 +
</table>

Revision as of 17:30, 16 October 2014

The algebra $k\langle X \rangle$ of polynomials over a field $k$ in non-commuting variables in $X$. The following universal property determines the algebra $k\langle X \rangle$ uniquely up to an isomorphism: There is a mapping $i : k \rightarrow k\langle X \rangle$ such that any mapping from $X$ into an associative algebra $A$ with a unit over $k$ can be factored through $k\langle X \rangle$ in a unique way. The basic properties of $k\langle X \rangle$ are:

1) $k\langle X \rangle$ can be imbedded in a skew-field (the Mal'tsev–Neumann theorem);

2) $k\langle X \rangle$ has a weak division algorithm, that is, the relation $$ d \left({ \sum_{i=1}^n a_i b_i }\right) < \max_i \{ d(a_i) + d(b_i) \} $$ where $a_i, b_i \in k\langle X \rangle$, all the $a_i$ are non-zero ($i = 1,\ldots,n$), $d(a_1) \le \cdots \le d(a_n)$, always implies that there are an integer $r$, $1 < r \le n$, and elements $c_,\ldots,c_{r-1}$ such that $$ d\left({ a_r - \sum_{i=1}^{r-1} a_i c_i }\right) < d(a_r) $$ and $$ d(a_i) + d(c_i) < d(a_r),\ \ \ i=1,\ldots,r-1 $$ (here $d(a)$ is the usual degree of a polynomial $a \in k\langle X \rangle$, $d(0) = -\infty$);

3) $k\langle X \rangle$ is a left (respectively, right) free ideal ring (that is, any left (respectively, right) ideal of $k\langle X \rangle$ is a free module of uniquely determined rank);

4) the centralizer of any non-scalar element of $k\langle X \rangle$ (that is, the set of elements that commute with a given element) is isomorphic to the algebra of polynomials over $k$ in a single variable (Bergman's theorem).

References

[1] P.M. Cohn, "Universal algebra" , Reidel (1981)
[2] P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
How to Cite This Entry:
Free associative algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_associative_algebra&oldid=16680
This article was adapted from an original article by L.A. Bokut (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article