Suppose $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is a short exact sequence of $R$-modules, where $R$ is a ring with unity.
If $A$ and $B$ are known, then $C$ is unique up to isomorphism, because by the definition of short exact sequences, $C \cong B/A,$ where $A$ is considered to be a submodule of $B$.
But what about the other cases?
Suppose $A$ and $C$ are known. What choices do we have for the module $B$? I know that $B$ is the direct sum of $A$ and $C$ if and only if the sequence at the top is a split exact sequence. But otherwise, is there anything we can say about the structure of $B$?
Or if $B$ and $C$ are known, what can we derive about the structure of $A$?
(I feel like there was a specific name for this type of problem which I once saw on Wikipedia, but have forgotten it, so it would be great if somebody could remind me of it!)