Definition 3

Let E and F be two (unital) A-algebras and M a set with an E-module structure and an F-module structure. M is called a (left) bimodule over the algebras E and F if:

(1) M is a bimodule over the underlying rings of E and F (II, § 1, no. 14); (2) the two A-module structures underlying the E-module and F-module structures on M are identical.

The latter condition says that if e and $e'$ are the unit elements of E and F respectively, then

then $\alpha x$ is used to denote the common value of the two sides.

It can also be said that being given on M a bimodule structure over E and F is equivalent to being given an A-module structure on M and also two A-algebra homomorphisms $s \mapsto h'_s$ of E into $\mathrm{End}_A(M)$ and $t \mapsto h''_t$ of F into $\mathrm{End}_A(M)$ such that $h'_s h''_t = h''_t h'_s$ for all $s \in E$ and $t \in F$. Consequently (no. 2, Proposition 5) an A-algebra homomorphism $u \mapsto h_u$ of $E \otimes_A F$ into $\mathrm{End}_A(M)$ is canonically derived such that $h_{s \otimes t} = h'_s h''_t = h''_t h'_s$ for $s \in E$ and $t \in F$. In other words, an $(E \otimes_A F)$-module structure is thus defined on M, which is said to be associated with the given bimodule structure over E and F and under which

The given E-module and F-module structures on M can be derived from this $(E \otimes_A F)$-module structure by restrictions of the ring of scalars, corresponding to the two canonical homomorphisms $E \to E \otimes_A F$ and $F \to E \otimes_A F$.

Conversely, if an $(E \otimes_A F)$-module structure is given on M, by means of the canonical homomorphisms $E \to E \otimes_A F$ and $F \to E \otimes_A F$ an E-module structure and an F-module structure on M and it is immediate that M is a bimodule over the algebras E and F with these two structures and that the given $(E \otimes_A F)$-module structure is associated with this bimodule structure.

Thus a one-to-one correspondence has been established between $(E \otimes_A F)$-module and bimodules over the algebras E and F. Clearly every sub-bimodule of M is a submodule for the associated $(E \otimes_A F)$-module structure and conversely. There are analogous results for quotients, products, direct sums and inverse and direct limits. Finally, if $M'$ is another bimodule over the algebras E and F and $f : M \to M'$ is a bimodule homomorphism, $f$ is also an $(E \otimes_A F)$-module homomorphism and conversely.

There are obviously corresponding statements for right bimodule structures, or when for example there is a left E-module structure and a right F-module structure; in this case we speak of an $(E, F)$-bimodule and being given such a structure amounts to being given a left bimodule structure over E and $F^o$.