Hi all,
Was wondering if I did my math right here. I have a battery 12v 4ah that I'm including in my chicken door circuit as a back up. I was hoping to charge it as well but don't want to over charge it. I am using a 1k resistor and two 12v 21ma zenar diodes. Here are a few pics of the circuit, battery and charger.

 

Is there something that I could add that will shut the charging voltage off after the battery is fully charged? Or should it matter?

 

You will only be charging the battery at 3mA and it will eventually go flat.

 

If the DC supply only outputs 12.0V it won't charge the battery.
A lead-acid battery's fully charged voltage is 12.6V, thus you need more than 13V to trickle-charge it (see here) and a higher voltage (at least 13.8V) to charge it.
Your best bet is likely to buy a smart charger designed for lead-acid batteries (such as these) and just leave that on the battery while you run the door circuit directly from the battery.