# Thread: Center of endomorphism algebra

1. ## Center of endomorphism algebra

If I have an R-algebra A where R is a ring, why the center of the endomorphism algebra $\displaystyle E_{A}(A)$ is the algebra of endomorphism with coefficient in the enveloping algebra $\displaystyle A* \otimes A$:

$\displaystyle Z(E_{A}(A))=E_{A* \otimes A}(A)$ ?

2. Originally Posted by yavanna87
If I have an R-algebra A where R is a ring, why the center of the endomorphism algebra $\displaystyle E_{A}(A)$ is the algebra of endomorphism with coefficient in the enveloping algebra $\displaystyle A* \otimes A$:

$\displaystyle Z(E_{A}(A))=E_{A* \otimes A}(A)$ ?
well, $\displaystyle End_A(A) \cong A$ and so you basically want to prove that $\displaystyle Z(A) \cong End_{A^* \otimes_R A}(A)$. this is a special case of a more general problem. for a solution to the general case see this theorem in my blog. the notation $\displaystyle C_A(B)$ means the centralizer of $\displaystyle B$ in $\displaystyle A$.

3. Thank you very much, i was trying to do that with the left regular representation right now, so I was on the right path!