Quotients of algebras; salvaging intuition One equation that every first-year student wants to be true is $A/B\cong C\implies A\cong B\times C$, where $A$, $B$, and $C$ are algebraic structures of some kind (modules, groups, rngs, . . .). However, this is not true: set $A=\mathbb{Z}$, $B=2\mathbb{Z}$, $C=\mathbb{Z}/2\mathbb{Z}$.
In the above case, though, $B\times C$ is not too far away from $A$: there is some $D$ such that $(B\times C)/D\cong A$. In this case, $D$ can be taken to be one of the copies of $B$.
My question is, when does this hold in general? One way (although certainly not the only way) to phrase this precisely would be to ask, what are those algebraic structures $A$ such that for all $B$, there is some $C$ such that $A\cong (B\times (A/B))/C$? (This is somewhat vague, as exactly what substructures $B$ are allowed here is not explicit, but I was thinking that this data might be incorporated into the data describing $A$; e.g., $A$ as a group, or $A$ as a ring without unity, etc.)
Thank you very much in advance!
 A: Another way to look at this is through exact sequences. Consider the exact sequence 
$$0 \longrightarrow B\longrightarrow A \longrightarrow C\longrightarrow 0.$$
You are asking when this thing splits. There are a bunch of equivalent conditions for exact sequences splitting. If these are modules, then certainly B being injective or C being projective will give it to you but those conditions are only sufficient, not necessary as far as I know. In the category of vector spaces over some fixed field, for example, the above sequence always splits because all vector spaces are free modules and hence projective. Note that the vector spaces don't even have to be finite dimensional. In this nice category, we even have that the dualizing functor (the functor sending a vector space to its dual, whatever you want to call that) is an exact functor (even though this fact may not be abundantly clear at first glance) which tells us that the dual of A is isomorphic to the product of the duals of B and C. I apologize for the lack of Tex. I'm gonna work on learning that. I would appreciate any Tex hints or advice anyone would be willing to give.
