Essential submodule

Remarks

Equivalently, for every nonzero $m \in M$ there is $r \in R$ with $rm \in N$ and $rm \ne 0$. Notation: $N \subseteq_e M$ or $N \trianglelefteq_e M$, depending on the author. Example: for $n \ne 0$, the submodule $n\Z$ is essential in the $\Z$-module $\Z$. The check: a nonzero submodule $a\Z$ of $\Z$ contains $na \ne 0$, which lies in $n\Z$. Non-example: $\Z \oplus 0$ is not essential in $\Z \oplus \Z$, since the nonzero submodule $0 \oplus \Z$ meets it only in $0$. A nonzero direct summand $N \ne M$ is never essential, since its complement meets it in $0$.