𝑅 is a field iff it has no nonzero proper ideals. So if 𝑅 is a field, dim⁡𝑅 =0. Conversely, if dim⁡𝑅 =0 then 0 is a maximal ideal (as Assuming choice, commutative ring has a maximal ideal and A maximal ideal in a commutative ring is prime), and thus 𝑅 has no nonzero proper ideals: 𝑅 is a field.

Note for dim⁡𝑅 =0 every nonzero prime ideal is maximal by vacuity. Given dim⁡𝑅 =1, then any nonzero prime ideal 𝔭 is contained within a maximal ideal 𝔪 which is also prime (since Every ideal in a commutative ring is contained in a maximal ideal), but this must be equal to 𝔭 or else 0 ⊊𝔭 ⊊𝔪 implies dim⁡𝑅 >1.