Let $R$ be a ring with unity and $S=\text{End}_R(M), {}_{S}M_R$ be an $(S,R)$-bimodule and $\{M_\lambda\}_{\lambda\in \Lambda}$ be a family of modules. Define the map $\phi:M\to M,\phi(m)= mr,r\in R)$ to be an endomorphism of $M$ for every $m\in M$.
Question (1): If the sequence $M_{\lambda}\stackrel{\cdot r}{\longrightarrow}M_{\lambda}\stackrel{\gamma}{\longrightarrow}N\longrightarrow0$ is exact for some $\gamma$, then show that the sequence $(\prod_{\lambda\in \Lambda}M_{\lambda})\stackrel{\cdot r}{\longrightarrow}(\prod_{\lambda\in \Lambda}M_{\lambda})\stackrel{\theta}{\longrightarrow}L\longrightarrow0$ is also exact for some $\theta\in \text{End}_R(M)$.
Question (2): Does the converse hold true? That is, if $ (\prod_{\lambda\in \Lambda}M_{\lambda})\stackrel{\cdot r}{\longrightarrow}(\prod_{\lambda\in \Lambda}M_{\lambda})\stackrel{\gamma}{\longrightarrow}L\longrightarrow0$ is exact for some $\gamma$, does that mean that $M_{\lambda}\stackrel{\cdot r}{\longrightarrow}M_{\lambda}\stackrel{\theta}{\longrightarrow}N\longrightarrow0$ is exact for some $\theta$?