Units of a finite-dimensional algebra
= Units of a finite-dimensional algebra
{title2=$E^\times$}
The unit group $E^\times$ is a nonempty open subset of the affine space $E$, detected by the determinant of left multiplication. It is therefore an irreducible connected <linear algebraic group>. Its map to $(E/J(E))^\times$ has kernel $1+J(E)$, a closed unipotent subgroup since the radical is nilpotent.