Not sure what you mean by "craziest" (most counterintuitive?), but I'll tell you a very important one that links algebra to geometry: in an arbitrary nonzero commutative ring, the intersection of all prime ideals is the set of nilpotent elements. So if an element of a nonzero commutative ring vanishes modulo all prime ideals, then that element might not be 0 but at least some power of it is 0.