Let $e$ be an idempotent of a ring $A$ and $N$ is an $A$-module. Why $\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(Ae,N), N\otimes_{eAe}A=N\otimes_{eAe}eA$? Can you prove this explicitly? Is the following true:
- $\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(eA,N)$
- $N\otimes_{eAe}A=N\otimes_{eAe}Ae$
- $\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(A,eN)$
- $N\otimes_{eAe}A=Ne\otimes_{eAe}A$
- $\mathrm{Hom}_{eAe}(A,N)=\mathrm{Hom}_{eAe}(A,Ne)$
- $N\otimes_{eAe}A=eN\otimes_{eAe}A$
Thank you.