View source for Unimodular element

''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" />.

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]].

====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>