elementary linear group functor

The elementary linear group functor is defined as functor

\begin{align*} \mathrm{E}_n: \mathrm{Ring} \rightarrow& \mathrm{Grp} \\ R \mapsto& \mathrm{E}_n(R) \\ (f: R \rightarrow S) \mapsto& (\mathrm{E}_n(f): \mathrm{E}_n(R) \rightarrow \mathrm{E}_n(S)) \end{align*}

where \(\mathrm{E}_n(f)\) is given by

\begin{align*} (\alpha_{i,j}) \mapsto (f(\alpha_{i,j})) \end{align*}

note that this is welldefined, as the imgae of an elementary matrix under \(f\) is again an elementary matrix.

see also:

• general linear group functor