On Wikipedia https://en.wikipedia.org/wiki/Isomorphism_theorems#Theorem_C_(modules), the second part of the third isomorphism theorem for modules says the following.
Let $M$ be an $R$ module and let $T$ be a submodule of $M$.
Then every submodule of $\frac{M}{T}$ is of the form $\frac{S}{T}$ for some submodule $S$ of $M$, such that $ T \subseteq S \subseteq M$.
Initially, I attempted as follows;
Let $I$ be a submodule of $\frac{M}{T}$. Define $\phi : I \rightarrow M$ by $\phi( m + T ) = m, m \in I $. Then I tried to use $\phi$ to show $T \subseteq Im(\phi) \subseteq M$ and $I = \frac{Im(\phi)}{T}$. However I then realised that $\phi$ is not well defined, since if $m_1 + T = m_2 + T$, both are in $I$, and $m_1 \neq m_2$, $\phi$ maps them to different elements even though they are the same element in the domain.
Does anyone have a hint/proof?
Thanks in advance
Best Answer
As answered in the comments, If $I$ is a submodule of $\frac{M}{T}$, the submodule of M given by $S = \{ m \in M : m + T \in I \} $ gives $ T \subseteq S \subseteq M$ and $I = \frac{S}{T} $.