You shouldn't store any cooked grain in the fridge for more than a week. This is particularly true of rye berries.
If you make a large batch and know you won't finish it all in a few days, a better option is to portion it and freeze it. It will reheat in the microwave in just a few minutes, or you can toss it in a sauce pan with some aromatics and veggies for a pilaf.
You can do the same thing with brown rice, millet, barley, any grain really. Just be careful when reheating it.