Vector space over a division ring ($V_{\Delta }\text{ or }{}_{\Delta }V$)

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 ${\mathrm{End}}_R(V{)}^{\mathrm{op}}$, a division ring by Schur's lemma. This is the scalar structure used by the Jacobson density theorem.