First function in a short exact sequence

Question

Let $0\to L\to M\to N \to0$ be short exact sequence. How does the function $0\to L$ look like? And what does $0$ mean here? Is it the zero of $A$ (a commutative ring s.t. $L,M,N$ are $A$-modules)?

Answer 1

Yes, if your sequence is one of $A$-modules, then the $0$ means, by slight abuse of notation the set containing only $0$, i.e. $\{0\}$, which is again an $A$-module. Furthermore you can only have one morphism out of this into any other $A$-module (because zero has to go to zero and that is all there is), so this is what it is. Similarly, the last arrow sends everything to zero.

Background information/terminology: If you like category theory language, $0$ is at the same time an initial and a terminal object in the category of modules, that means to every other module there exists exactly one morphism from $0$ (initial) and out of every other module there is exactly one morphism to $0$ (terminal). Such objects are called (probably because of this example) zero objects.