Are direct products of simple modules co-semisimple?

Question

Let $M$ be an $R$-module for a unital ring $R$ and $M=\prod_{\alpha\in A}M_\alpha$ where $M_\alpha$'s are all simple $R$-modules. My question is:

For any proper submodule $K$ of $M$ and any $x\in M\backslash K$, is there always a maximal submodule $N$ of $M$(that is, $M$ is maximal among all proper submodule of $M$) such that $K\leqslant N$ and $x\notin N$?

I've noticed that the property in consideration is equivalent to the following two:

a) $M$ is co-semisimple, that is, every submodule of $M$ is a intersection of a set of maximal modules of $M$.

b) for any proper submodule $K$ of $M$ and any $x\in M\backslash K$, we can always find a proper submodule $N$ of $M$ such that $K\leqslant N$ and $Rx+N=M$.

Answer 1

Let $R=\mathbb{Z}$ and $M=\prod_p\mathbb{Z}/p\mathbb{Z}$, the product being over all primes $p$.

Any maximal submodule $N$ of $M$ is the kernel of a map to $\mathbb{Z}/q\mathbb{Z}$ for some prime $q$, so $N$ contains $qM=\prod_{p\neq q}\mathbb{Z}/p\mathbb{Z}$. Since $qM$ is maximal, $N=qM$. In other words, the only maximal submodules of $M$ are the obvious ones.

So if $K=\bigoplus_p\mathbb{Z}/p\mathbb{Z}$, then $K$ is not contained in any maximal submodules of $M$.