Exercise 15, Appendix 1

Let A be a $k$-algebra. A multiplier of A is a multiplier of the $(A,A)$-pseudobimodule A. We denote the $k$-algebra of multipliers of A by $\mathscr{M}(A)$. a) Prove that the mapping $\mu:a\mapsto m_a$ is a $k$-algebra homomorphism and that we have $am=m_am$ and $ma=mm_a$ for $a\in A, m\in \mathscr{M}(A)$. The image of the homomorphism $\mu$ is therefore a two-sided ideal of the ring $\mathscr{M}(A)$.

b) A two-sided ideal $\mathfrak{b}$ of a $k$-algebra A is called faithful if its left and right annihilators are zero. Show that if A is faithful in A, then the homomorphism $\mu$ is injective and the ideal $\mu(A)$ is faithful in $\mathscr{M}(A)$.

c) Let B be a unital $k$-algebra containing A as a two-sided ideal. For $b\in B$, let $\sigma_b$ and $\delta_b$ be the mappings from A to A such that for $a\in A$, we have $\sigma_b(a) =ba$ and $\delta_b(a) =ab$, and set $\pi (b) = (\sigma_b, \delta_b)$. Show that $\pi (b)\in \mathscr{M}(A)$ for every $b\in B$ and that the resulting mapping $\pi : B\rightarrow \mathscr{M}(A)$ is a ring homomorphism whose restriction to A is the mapping $a\mapsto m_a$. Show that $\pi$ is injective if the ideal A is faithful in B.