Let A {\displaystyle {}A} be a commutative R {\displaystyle {}R} -algebra over a commutative ring R {\displaystyle {}R} . Let
denote the R {\displaystyle {}R} -linear multiplication map for f ∈ A {\displaystyle {}f\in A} . For R {\displaystyle {}R} -linear maps
set
Suppose that a R {\displaystyle {}R} -derivation δ : A → A {\displaystyle {}\delta \colon A\rightarrow A} is given. Show that for all g ∈ A {\displaystyle {}g\in A} the map [ δ , μ g ] {\displaystyle {}[\delta ,\mu _{g}]} is multiplication by some element.
be a commutative 
-algebra over a commutative ring . Let
-linear multiplication map for
.
For -linear maps
![{\displaystyle {}[\varphi _{1},\varphi _{2}]=\varphi _{1}\circ \varphi _{2}-\varphi _{2}\circ \varphi _{1}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7307941213194905b6c44144d9513a4ea67850b3)
-derivation
is given. Show that for all
the map ![{\displaystyle {}[\delta ,\mu _{g}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5bd0329f0a6f24937bddc092edbdc7758a6d9f)
is multiplication by some element.
