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?