Is $2\Bbb Z_4$ a summand of $\Bbb Z_4$? I figured the answer would be no since there are no submods of $\Bbb Z_4$ that do not intersect with $2\Bbb Z_4$, but I am not sure.

The only nonzero proper submodule of $\mathbb Z_4$ is $2 \mathbb Z_4$. For $2\mathbb Z_4$ to be a summand, there would need to be another nonzero proper submodule. Since such a submodules does not exist, $2\mathbb Z_4$ cannot be a summand.
