= Vector space over a division ring
{title2=$V_\Delta\text{ or }{}_\Delta V$}
A <module> over a <division ring> has the usual <basis> and <linear independence> theory of a <vector space>, with the side of scalar multiplication specified. For a right <module>, a finite relation has the form $\sum_i v_i a_i=0$. Every nonzero <endomorphism> of a one-dimensional right <module> is invertible. If $V$ is a <simple module> on the left over $R$, evaluation makes it a right vector space over the <opposite ring> $\operatorname{End}_R(V)^{\mathrm{op}}$, a <division ring> by <Schur's lemma>. This is the scalar structure used by the <Jacobson density theorem>.
Back to article page
