Example

(1) On a ring A the module structures of $A_s$ and $A_d$ are compatible and A can therefore be considered canonically as an (A, A)-bimodule.

(2) A left A-module E has a canonical left module structure over the ring $\operatorname{End}_A(E)$ and the A-module and $\operatorname{End}_A(E)$-module structures on E are compatible.

Clearly when E is a multimodule over two families $(A_\lambda)_{\lambda \in L}, (B_\mu)_{\mu \in M}$ of rings, E is also a multimodule over any two subfamilies $(A_\lambda)_{\lambda \in L'}, (B_\mu)_{\mu \in M'}$, where the $A_\lambda$-module and $B_\mu$-module structures for $\lambda \in L'$ and $\mu \in M'$ are those initially given.

Since multimodules are particular examples of commutative groups with operators, the results of nos. 2 to 10 (cf. no. 10, Remark) can be applied to them; in particular, if E, F are two $((A_\lambda), (B_\mu))$-multimodules, a homomorphism $u : E \to F$ is a mapping which is an $A_\lambda$-homomorphism for all $\lambda \in L$ and a $B_\mu$-homomorphism for all $\mu \in M$. The stable subgroups of an $((A_\lambda), (B_\mu))$-multimodule are $((A_\lambda), (B_\mu))$-multimodules (called submultimodules), as also are the quotients by such subgroups (called quotient multimodules); similarly for products and direct sums.

Let E be an $((A_\lambda), (B_\mu))$-multimodule and for each $\lambda \in L$ (resp. $\mu \in M$) let $\phi_\lambda : A'_\lambda \to A_\lambda$ (resp. $\psi_\mu : B'_\mu \to B_\mu$) be a ring homomorphism; clearly the $A'_\lambda$-module structures associated with the $\phi_\lambda$ and the $A_\lambda$-module structures given on E and the $B'_\mu$-module structures associated with the $\psi_\mu$ and the $B_\lambda$-module structures given on E are compatible with one another and hence define on E an $((A'_\lambda), (B'_\mu))$-multimodule structure, said to be associated with the given $((A_\lambda), (B_\mu))$-multimodule structure and the $\phi_\lambda$ and $\psi_\mu$.

If E, F are two $((A_\lambda), (B_\mu))$-multimodules, the additive group of homomorphisms of E into F is denoted by $\operatorname{Hom}_{(A_\lambda), (B_\mu)}(E, F)$ (or simply $\operatorname{Hom}(E, F)$). Formulae (6) to (8) of no. 2 are obviously valid for $((A_\lambda), (B_\mu))$-multimodule homomorphisms and, in particular, $\operatorname{Hom}(E, E) = \operatorname{End}(E)$ has a ring structure;

moreover Hom(E, F) has a canonical left End(F)-module structure and right End(E)-module structure, these two structures being compatible; in other words, Hom(E, F) has a canonical (End(F), End(E))-bimodule structure.

Suppose now that E has a multimodule structure whose rings of left (resp. right) operators are on the one hand the $A_\lambda$ for $\lambda \in L$ (resp. the $B_\mu$ for $\mu \in M$) and on the other the rings of another family $(A'_{\lambda'})_{\lambda' \in L'}$ (resp. $(B'_{\mu'})_{\mu' \in M'}$). Suppose similarly that F has a multimodule structure whose rings of left (resp. right) operators are on the one hand the $A_\lambda$ for $\lambda \in L$ (resp. the $B_\mu$ for $\mu \in M$) and on the other the rings of another family $(A''_{\lambda''})_{\lambda'' \in L''}$ (resp. $(B''_{\mu''})_{\mu'' \in M''}$); to abbreviate we shall say that E is an (($A_\lambda$), ($A'_{\lambda'}$); ($B_\mu$), ($B'_{\mu'}$))-multimodule and F an (($A_\lambda$), ($A''_{\lambda''}$); ($B_\mu$), ($B''_{\mu''}$))-multimodule. Consider E and F as (($A_\lambda$), ($B_\mu$))-multimodules, thus restricting the operators to the subfamilies ($A_\lambda$) and ($B_\mu$). By what was said at the beginnings of this no., the multimodule structures given on E and F define canonically ring homomorphisms

moreover, two elements of $\operatorname{End}_{(A_\lambda), (B_\mu)}(E)$ (resp. $\operatorname{End}_{(A_\lambda), (B_\mu)}(F)$) respective images of elements of two distinct rings among the $A'_{\lambda'}$, or the ${B'_{\mu'}}^0$ (resp. the $A''_{\lambda''}$ or the $B''_{\mu''}$) are permutable; it follows that the above homomorphisms define on $\operatorname{Hom}_{(A_\lambda), (B_\mu)}(E, F)$ a multimodule structure whose rings of left operators are the $A''_{\lambda''}$ ($\lambda'' \in L''$) and the $B'_{\mu'}$ ($\mu' \in M'$) and whose rings of right operators are the $A'_{\lambda'}$ ($\lambda' \in L'$) and the $B''_{\mu''}$ ($\mu'' \in M''$).

If now $E'$ is an (($A_\lambda$), ($A'_{\lambda'}$); ($B_\mu$), ($B'_{\mu'}$))-multimodule and $F'$ an (($A_\lambda$), ($A''_{\lambda''}$); ($B_\mu$), ($B''_{\mu''}$))-multimodule, $\operatorname{Hom}_{(A_\lambda), (B_\mu)}(E', F')$ is an

if $u : E' \to E, v : F \to F'$ are multimodule homomorphisms,

is defined as in no. 2 and is a multimodule homomorphism.