Proposition 1

a) The mapping $\varphi$ induces a bijection from the set of left (resp. right, two-sided) ideals of A of finite codimension to the set of right A-submodules (resp. left A-submodules, sub-bimodules) of $A^*$ of finite dimension over K.

b) Every left or right ideal of A of finite codimension contains a two-sided ideal of finite codimension.

Formulas (3) and (4) prove that if E is a left ideal of A, then $E'$ is a right A-submodule of $A^*$; if V is a right A-submodule of $A^*$, then $V'$ is a left ideal of A. The cases of right ideals and of two-sided ideals are treated analogously. This proves a).

Let $\mathfrak{a}$ be a left ideal of A of finite codimension. The left A-module $E = A_s/\mathfrak{a}$ is finite-dimensional over K, and so is End$_K(E)$. The kernel of the homomorphism $a\mapsto a_E$ from A to End$_K(E)$ is therefore a two-sided ideal of A of finite codimension contained in $\mathfrak{a}$. Passing to the opposite ring $A^o$, we see that every right ideal of A of finite codimension contains a two-sided ideal of finite codimension.