Difference between revisions of "Unimodular element"
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 17:35, 31 March 2012
unimodular vector
Let be a ring with unit and
a right module over
. An element
in
is called unimodular if
and the submodule
generated by
has a complement
in
, i.e. there is a submodule
such that
,
, so that
.
An element of a free module that is part of a basis of
is unimodular. An element
is unimodular if and only if there is a homomorphism of modules
such that
. A row (or column) of a unimodular matrix over
is unimodular. The question when the converse is true is important in algebraic
-theory. Cf. also Stable rank.
References
[a1] | A.J. Hahn, O.T. O'Meara, "The classical groups and ![]() |
How to Cite This Entry:
Unimodular element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_element&oldid=17441
Unimodular element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_element&oldid=17441