Prove that a commutative ring F is a field iff each equation ax+b=c
(a,b,c is a member of F and a does not equal 0) has a unique solution in F
well, the condition you have is clearly equaivalent to say that has a unique solution for any if is a field and then from you'll get and you're done.
but the converse is less trivial: so we have that any equation has a unique solution in first we show that has no zero divisor: suppose that and now the
equation has two solutions thus this proves that has no zero divisor. next we show that is unitary: choose any then for some now
let then there exists such that but then since has no zero divisor, we must have clearly thus: therefore
finally for any the equation has a (unique) solution, which means any non-zero element of is invertible.