Definition:-1 A submodule $N$ of a module $M$ is said to be superfluous (or small) if there is no proper submodule $K$ of $M$ such that $M=N+K$.
Definition:-2 Jacobson radical $J(M)$ of a module $M$ is the sum of all superfluous submodules of $M$.
I have a confusion about the trivial case. Clearly, zero submodule of every nonzero module is superfluous. But when $M=0$ then $J(M) =0$. From these notions, i think $0$ should be a superfluous submodule of $0$. Please clear my doubt. I will be highly thankful to You.
$0$ is also a superfluous submodule of $M=0$, because for all proper submodules $K\subsetneq M$, we have $0+K\ne 0$.