Maximal submodules and maximal ideals

Question

Let $R$ be a commutative ring and $M$ be an $R$-module. We know that $Rad(M)$ is the intersection of all maximal submodules of $M$. If $K$ is a maximal ideal of $R$, is it true that $Rad(M)$ is necessarily a subset of $KM$?

Answer 1

Note that $M/KM$ is a $R/K$-vector space ($R/K$ is a field as $K$ is maximal). Hence $Rad(M/KM)$ vanishes (this follows from linear algebra). By the correspondence between submodules of $M/KM$ and submodules of $M$ containing $KM$ it follows that $KM$ is the intersection of the maximal submodules containing it. Thus $Rad(M)$ is contained in $KM$.