Exercise 36, § 7

(a) Let $K_0$ be a commutative field and $K = K_0(X)$ the field of rational functions in one indeterminate over $K_0$ (Chapter IV). A $(K, K)$-bimodule structure is defined on $K$ as follows: the product $t.u$ of an element $u \in K$ by a left operator $t \in K$ is the rational function $t(X)u(X)$; the product $u.t$ of $u$ by a right operator $t \in K$ is the rational function $u(X)t(X^2)$; the left (resp. right) vector $K$-space structure thus defined on $K$ is then 1-dimensional (resp. 2-dimensional). Deduce examples of $(K, K)$-bimodules such that dimensions of the two vector space structures of such a bimodule are arbitrary integers.

(b) Deduce from (a) an example of a $(K, K)$-bimodule $E$ whose two vector space structures have the same dimension and such that there exists a sub-bimodule $F$ of $E$ whose two vector space structures do not have the same dimension.*