Exercise 35, § 7

(a) Let $K$ be a field and $E$ a $(K, K)$-bimodule. Suppose that the dimensions of $E$, as a left and right vector space over $K$, are equal to the same finite number $n$. Show that there exists a family $(e_i)$ of $n$ elements of $E$, which is a basis for each of the two vector space structures on $E$. (Note that if $(b_j)_{1 \leq j \leq m}$ is a family of $m < n$ elements of $E$, which is free for each of the two vector space structures on $E$ and $V$ is the left vector subspace and $W$ the right vector subspace generated by $(b_j)$, either $V + W \neq E$, or $V \cap CW$ and $W \cap CV$ are non-empty; in the latter case, if $y \in V \cap CW, z \in W \cap CV$, $y + z$ forms with the $b_j$ a family of $m + 1$ elements which is free for each of the two vector space structures on $E$.)

(b) Let $F$ be a subbimodule of $E$, whose two vector space structures over $K$ have the same dimension $p < n$. Show that there exists a family $(e_i)_{1 \leq i \leq n}$ of elements of $E$ which is a basis of $E$ for each of its vector space structures and such that $(e_i)_{1 \leq i \leq p}$ is a basis of $F$ for each of its two vector space structures (same method).

(c) Let $(b_j)_{1 \leq j \leq n-1}$ be a family of $n - 1$ elements of $E$, which is free for each of the two vector space structures on $E$. Let $V$ (resp. $W$) be the hyperplane generated by the $(b_j)$ in $E$ considered as a left (resp. right) vector space.

Show that if $V \subset W$ (resp. $W \subset V$), then necessarily $V = W$ (note that if $a \notin W$, the set of $\lambda \in K$ such that $\lambda a \in W$ is a left ideal).