$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\End{End}$Let $M_1$, $M_2$ be a $R$-modules and $M := M_1 \oplus M_2$ if $ \pi_1$, $\pi_2$ are the respectives projections and any $ \phi \in \End_R(M)$ has a form $$ \phi = \sum_{i = 1}^{2} \sum_{j = 1}^{2} \pi_i \phi \pi_j $$ We define $\phi_{ij} = \pi_i \phi \pi_j|_{M_j}$, I need to show that the morphism $$ \rho: \End_R(M) \to {M_{2\times2}(\Hom_R(M_j, M_i))}$$ given by $\rho(\phi) = \begin{bmatrix} \phi_{11} & \phi_{12} \\ \phi_{21} & \phi_{22} \\ \end{bmatrix}$
is a isomorphism to rings.
I showed $\rho( \phi + \psi) = \rho( \phi) + \rho(\psi)$ and also showed that is surjectivity and injectivity buy i not can show that $\rho (\phi \circ \psi) = \rho (\phi) \rho(\psi) $, I try the direct calculus but apparently it is not the way.
some help?
thank
Your definition of $\phi_{ij}$ is wrong. Let $\varkappa_j:M_j\to M$ be the inclusion morphism into the direct sum and define $\phi_{ij}=\pi_i\circ\phi\circ\varkappa_j:M_j\to M_i$. Then we have $$\begin{bmatrix} \phi_{11}&\phi_{12}\\ \phi_{21}&\phi_{22} \end{bmatrix} \begin{bmatrix} \psi_{11}&\psi_{12}\\ \psi_{21}&\psi_{22} \end{bmatrix}= \begin{bmatrix} \phi_{11}\circ\psi_{11}+\phi_{12}\circ\psi_{21}&\phi_{11}\circ\psi_{12}+\phi_{12}\circ\psi_{22}\\ \phi_{21}\circ\psi_{11}+\phi_{22}\circ\psi_{21}&\phi_{21}\circ\psi_{12}+\phi_{22}\circ\psi_{22} \end{bmatrix}$$ but since $\varkappa_1\circ\pi_1+\varkappa_2\circ\pi_2=\mathrm{id}_M$, we get $$\begin{bmatrix} \phi_{11}&\phi_{12}\\ \phi_{21}&\phi_{22} \end{bmatrix} \begin{bmatrix} \psi_{11}&\psi_{12}\\ \psi_{21}&\psi_{22} \end{bmatrix}= \begin{bmatrix} (\phi\circ\psi)_{11}&(\phi\circ\psi)_{12}\\ (\phi\circ\psi)_{21}&(\phi\circ\psi)_{22} \end{bmatrix}$$ for \begin{align} \phi_{i1}\circ\psi_{1j}+\phi_{i2}\circ\psi_{2j} &=\pi_i\circ\phi\circ\varkappa_1\circ\pi_1\circ\psi\circ\varkappa_j+\pi_i\circ\phi\circ\varkappa_2\circ\pi_2\circ\psi\circ\varkappa_j\\ &=\pi_i\circ\phi\circ(\varkappa_1\circ\pi_1+\varkappa_2\circ\pi_2)\circ\psi\circ\varkappa_j\\ &=\pi_i\circ\phi\circ\psi\circ\varkappa_j\\ &=(\phi\circ\psi)_{ij} \end{align}