I breed in my apartment with no problems at all. I have 5 females, 2 males, and a few misc ones just for fun.
I have an air purifier, and it works magic! Keeps the smell and rat dander down. Also provides air flow. So I highly recommend a purifier.
You can also litter train your rats. They're smart animals who are very capable of learning. Litter training keeps messes to a minimum. And dumping a litter box beats dumping a cage or tub.
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