Battery of ups/inverter

I have inverter at my home but my inverter will restart my computer when the electricity goes down. I want to buy UPS because the UPS will not restart the desktop computer. I want to know that one can use the same battery for UPS and inverter or the battery of UPS is different than the Battery of inverter? There is difference in the price of the battery of UPS and inverter or one can use same battery for both?

you can use same battery for both ups/inverter.

Same type battery can be used with Inverter/UPS/Vehicle etc.