# Unimodular element

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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.

How to Cite This Entry:
Unimodular element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unimodular_element&oldid=49077