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)?
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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.
jorst
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Examples: In the category of abelien (or general) groups, $0$ is the trivial group ("the" is somewhat justified because we have uniqueness up to unique isomorphism); in the category of $k$-vector spacs it is the zero-dimensional space; in the category of sets, there is no zero (which is only one reason why we cannot talk about exact sequences there): $\emptyset$ is initial and a singleton set is terminal. – Hagen von Eitzen Oct 18 '14 at 11:51