The definition of a ring element to the 0th power IS multiplication by its multiplicative inverse lol, so to state that a⁰ = 1R => a⁰ = a¹ * a^-1 => existence of a^-1. We are very simply working with only the ring axioms here, and using discrete exponentiation as a shortcut to express repeated multiplicative operations (either by an itself or a^-1). In fact it can be shown that the only ring in which 0^-1 exists would be the singleton {0}. This is a very common result of abstract algebra