Exact functors and "short" exact functors Let $A$ be a commutative ring. I thought that an exact functor (from the category of $A$-modules to itself) is defined to be a functor which sends every exact sequence to an exact sequence. But many books seem to define it to be a functor which sends every short exact sequence to a short exact sequence. But I don't think they are equivalent unless it sends a zero module to a zero module. 

Is it true that every functor on the category of $A$-modules to itself sends a zero module to a zero module?


The "correct" definitions (in that they generalize the most smoothly to non-additive contexts) are that a functor is 


*

*left exact if it preserves finite limits, 

*right exact if it preserves finite colimits, and 

*exact if it preserves both. 


For a functor between abelian categories, the last condition is equivalent to being additive and preserving short exact sequences (exercise). In any case, by default people usually only consider additive functors between abelian categories anyway. 
