Exercise 14, Appendix 1

Let A and B be $k$-algebras; an $(A,B)$-pseudobimodule is a set E endowed with the structure of a left pseudomodule over A and the structure of a right pseudomodule over B such that the underlying $k$-module structures coincide. This corresponds to endowing E with the structure of a bimodule over the algebras $\widetilde{A}$ and $\widetilde{B}$. A multiplier of E is a pair $(S,D)$, where $S : B\rightarrow E$ is a homomorphism of right B-pseudomodules and $D : A\rightarrow E$ is a homomorphism of left A-pseudomodules, satisfying $aS(b) = D(a)b$ for $a\in A,b\in B$. We denote by $\mathscr{M}(E)$ the set of multipliers of E; it admits a natural $k$-module structure.

a) For every $x\in E$, we denote by $D_x$ (resp. $S_x$) the mapping $a\mapsto ax$ (resp. $b\mapsto xb$). Show that the pair $m_x= (D_x,S_x)$ is a multiplier of E.

b) For every $m= (S,D)\in \mathscr{M}$(E), $a\in A$, and $b\in B$, set $am=m_{D(a)}$ and $mb=m_{S(b)}$. Prove that this defines on the $k$-module $\mathscr{M}(E)$ the structure of a $(A$, B)-pseudobimodule and that the mapping $x\mapsto m_x$ is a homomorphism of $(A$, B)-pseudobimodules.