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''unimodular vector''
 
''unimodular vector''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953401.png" /> be a ring with unit and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953402.png" /> a right module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953403.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953404.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953405.png" /> is called unimodular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953406.png" /> and the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953407.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953408.png" /> has a complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u0953409.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534010.png" />, i.e. there is a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534013.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534014.png" />.
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Let $  R $
 +
be a ring with unit and $  M $
 +
a right module over $  R $.  
 +
An element $  x $
 +
in $  M $
 +
is called unimodular if $  \mathop{\rm ann} _ {R} ( x) = \{ {r \in R } : {xr = 0 } \} = 0 $
 +
and the submodule $  \langle  x \rangle $
 +
generated by $  x $
 +
has a complement $  N $
 +
in $  M $,  
 +
i.e. there is a submodule $  N \subset  M $
 +
such that $  \langle  x \rangle \cap N = \{ 0 \} $,  
 +
$  \langle  x \rangle + N = M $,  
 +
so that $  \langle  x \rangle \oplus N = M $.
  
An element of a free module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534015.png" /> that is part of a basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534016.png" /> is unimodular. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534017.png" /> is unimodular if and only if there is a homomorphism of modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534019.png" />. A row (or column) of a [[Unimodular matrix|unimodular matrix]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534020.png" /> is unimodular. The question when the converse is true is important in [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534021.png" />-theory]]. Cf. also [[Stable rank|Stable rank]].
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An element of a free module $  M $
 +
that is part of a basis of $  M $
 +
is unimodular. An element $  x \in M $
 +
is unimodular if and only if there is a homomorphism of modules $  \rho : M \rightarrow R $
 +
such that $  \rho ( x) = 1 $.  
 +
A row (or column) of a [[Unimodular matrix|unimodular matrix]] over $  R $
 +
is unimodular. The question when the converse is true is important in [[Algebraic K-theory|algebraic $  K $-
 +
theory]]. Cf. also [[Stable rank|Stable rank]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.J. Hahn, O.T. O'Meara, "The classical groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534022.png" />-theory" , Springer (1989) pp. 9, §141ff {{MR|1007302}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.J. Hahn, O.T. O'Meara, "The classical groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095340/u09534022.png" />-theory" , Springer (1989) pp. 9, §141ff {{MR|1007302}} {{ZBL|}} </TD></TR></table>

Latest revision as of 08:27, 6 June 2020


unimodular vector

Let $ R $ be a ring with unit and $ M $ a right module over $ R $. An element $ x $ in $ M $ is called unimodular if $ \mathop{\rm ann} _ {R} ( x) = \{ {r \in R } : {xr = 0 } \} = 0 $ and the submodule $ \langle x \rangle $ generated by $ x $ has a complement $ N $ in $ M $, i.e. there is a submodule $ N \subset M $ such that $ \langle x \rangle \cap N = \{ 0 \} $, $ \langle x \rangle + N = M $, so that $ \langle x \rangle \oplus N = M $.

An element of a free module $ M $ that is part of a basis of $ M $ is unimodular. An element $ x \in M $ is unimodular if and only if there is a homomorphism of modules $ \rho : M \rightarrow R $ such that $ \rho ( x) = 1 $. A row (or column) of a unimodular matrix over $ R $ is unimodular. The question when the converse is true is important in algebraic $ K $- theory. Cf. also Stable rank.

References

[a1] A.J. Hahn, O.T. O'Meara, "The classical groups and -theory" , Springer (1989) pp. 9, §141ff MR1007302
How to Cite This Entry:
Unimodular element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_element&oldid=49077