I need some help with this exercise (maybe just for the first part, or something to begin with and then I will try too solve the rest).
Let $R$ be a commutative ring with identity. Let $I\subset R$ an ideal and $M$ a finitely generated $R$-Module. Then:
i) If $T: M\rightarrow M$ is an R-Modules homomorphism such that $T(M)\subset I\cdot M$, then exist a monic polynomial $p$ with coefficients in $I$ such that $p(T)=0$.
ii)If $I\cdot M=M$ then, exist $r\in R$ such that $r-1\in I$ and $r\cdot M=0$.