Let $(M,*)$ be a monoid. $(N,*)$ is a submonoid if it is a monoid and $N\subseteq M$.
A submonoid does not always inherit the neutral element.
Example.$(\N,\max)$ is a monoid with neutral element $0$, but $(\N^+,\max)$ is a submonoid with neutral element $1$.
Let $(M,*)$ be a monoid. $N\subseteq M$ is a submonoid as soon as:
Closure. For all $a$, $b\in N$,
\[ab \in N.\]Neutral element. There is $e\in N$ such that for all $a\in N$,
\[ae = ea = a.\]