Example

(1) Let B be an A-algebra; the ring B has canonically a (B, B)-bimodule structure (II, § 1, no. 14, Example 1) and, if e is the unit element of B, then $(\alpha e)x = x(\alpha e) = \alpha x$ for all $x \in B$ and all $\alpha \in A$; B can therefore be considered as a left bimodule over the algebras B and $B^o$ (opposite to B); there is therefore associated with the (B, B)-bimodule structure on B a (B $\otimes_A B^o$)-module structure such that, for $b, x$ and $b'$ in B,

the right-hand side being the product in the ring B.

(2) Let E and F be two A-algebras, $e, e'$ their respective unit elements, M an E-module and N an F-module; these module structures define on M a bimodule structure over the rings A and E and on N a bimodule structure over the rings A and F; from these is therefore derived a bimodule structure over the rings E and F on the tensor product $M \otimes_A N$, defined by

for $x \in E, y \in F, m \in M, n \in N$ (II, § 3, no. 4); it is also seen that conditions (8) hold and hence the above bimodule structure is associated with an $(E \otimes_A F)$-module structure on $M \otimes_A N$, such that

for $x \in E, y \in F, m \in M, n \in N$.

In particular, taking $M = E_s, E_s \otimes_A N$ has canonically an $(E \otimes_A F)$-module structure; on the other hand, $E \otimes_A N$ is canonically identified with $E \otimes_A (F_d \otimes_F N) = (E \otimes_A F) \otimes_F N$, where $E \otimes_A F$ is considered as having its right F-module structure defined by the canonical homomorphism $v : F \to E \otimes_A F$; for $x, x'$ in E, $y \in F, n \in N$, $x' \otimes n$ is thus identified with $(x' \otimes e') \otimes n$ and $(x \otimes y).(x' \otimes n') = (xx') \otimes (y.n)$ with $((xx') \otimes y) \otimes n$. The $(E \otimes_A F)$-module $E_s \otimes_A N$ is thus identified with the $(E \otimes_A F)$-module derived from N by extending the scalars to $E \otimes_A F$ by means of the homomorphism $v$ (II, § 5, no. 1). The canonical mapping $n \mapsto e \otimes n$ of N into $E_s \otimes_A N$ is identified with the canonical mapping $n \mapsto (e \otimes e') \otimes n$ of N into $(E \otimes_A F) \otimes_F N$; this is known to be an F-homomorphism.

With the same notation, let P be a right (E \otimes_A F)-module; then there is a canonical Z-module isomorphism

(11)

where on the right-hand side P is considered as a right F-module by means of the canonical homomorphism $v$. For P is canonically identified with $P \otimes_{E \otimes_A F} (E \otimes_A F)$ and $(E \otimes_A F) \otimes_F N$ with $E \otimes_A (F \otimes_F N)$ and hence with $E \otimes_A N$, which establishes the stated isomorphism (II, § 3, no. 8, Proposition 8 and II, § 3, no. 4, proposition 4).

All the above extends to multimodules (II, § 1, no. 14).