Do you already have a Z-Wave system? If not, there may be easier solutions.
I don't know anything about that module, but you'd most likely need a relay & relay driver (or a relay board with a driver built-in). It looks like it works as a receiver, but I'm not sure if it's a transmitter, receiver, or both.
In any case, you'll have to power the thing and a couple of AA batteries probably won't do it. And, the receiver has to be powered-up in order to receive an on-signal. It might be easier to get a power supply/DC adapter and plug into a Z-Wave controlled outlet.
I also got a couple of hits Googling "Z Wave relay" or "Z-Wave dry contact" (but, nothing looked "cheap").