Torsion submodules

Suppose $R$ is a ring and $M$ is a left $R$-module. An element $m\in M$ is called a torsion element if $rm=0$ for some nonzero $r\in R$. The set of torsion elements in $M$ is denoted $\mathrm{Tor}(M)$.

1. Prove that if $R$ is an integral domain then $\mathrm{Tor}(M)$ is a submodule of $M$.
2. Give an example of a ring $R$ and $R$-module $M$ such that $\mathrm{Tor}(M)$ is not a submodule. (Hint: Consider torsion elements in the $R$-module $R$ for some specific ring $R$.)
3. Suppose $R$ has a (nonzero) zero divisor and $M$ is nontrivial. Prove that $M$ has nonzero torsion elements.
4. Suppose $\varphi :M\to N$ is an $R$-module morphism. Prove that $\varphi (\mathrm{Tor}(M))\subseteq \mathrm{Tor}(N)$.