Let \( M \) be an \( A \)-module and let \( N \) and \( P \) be submodules of \( M \). Show that \[ \displaystyle ( N \, \colon P ) \; \; \colon = \left\{ a \in A \; \Big | \; aP \subseteq N \right\} \]is an ideal of \( A \). If we set \( N=0 \) and \( P=M \), then we get the annihilator \( Ann(M) \) of \( M \). If \( I \) is an ideal of \( A \) such that \( I \subseteq Ann(M) \), then show that \( M \) has an \( A / I \)-module structure.
Additionally, prove the following:
- \( Ann(M+N)=Ann(M) \cap Ann(N) \)
- \( ( N \, \colon P ) = Ann( (N+P)/P ) \)
The post was edited, because it contained a mistake (Instead of \( M \) I had written \( Ann(M) \) in the second part - now the correct statement is marked in red)