In this question all rings are commutative, but don't necessarily have a multiplicative identity (so: commutative rngs). On Wikipedia there is the unsourced claim:
On the page this appears, von Neumann regular rings are allowed to be non-unital. However on the page dedicated to von Neumann regular rings, the definition states up front the assumption the ring is unital. So I am willing to accept (though a reference would be nice) that a unital commutative von Neumann regular local ring is a field. But I can define a local ring to be a ring with a unique maximal ideal, with no reference to the multiplicative identity. So my question is:
Is a commutative von Neumann regular ring with a unique maximal ideal necessarily a field, even if I don't assume the ring unital a priori?
On the one hand, this seems rather strong: where does the unit come from? But having a unique maximal ideal is rather strong, so perhaps this is enough.