Problem(1.1)

Let $V$ be an $A$-module. Show that $V$ is completely reducible iff the intersection of all of the maximal submodules of $V$ is trivial. However, this is not valid for the regular module of the ring of integers $\mathbb{Z}$.

Pf:

The necessity is obviously. Now we consider the sufficiency:

• $\cap_{i=1}^k M_i=0$ where the $M_i$ are maximal submodules of $V$.
• $N_i=\sum_{i\neq j}M_j$ where the $N_i$ is the irreducible submodule of $V$
• $V=M_i\oplus N_i$, then we have $V=V/(\cap M_i)\leq\oplus_i V/{M_i}=\oplus_i N_i$
• Show $\sum N_i=\oplus N_i$ by definition
• About the counterexample: $\cap_{p}(p)=0$, but if $\mathbb{Z}=(2)\oplus N$, then there exists an even number $0\neq k\in (2)\cap N$, a contradiction.