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

(Thinking)

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- Dec 7th 2008, 05:22 PMmandy123Prove it is a field
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

(Thinking) - Dec 7th 2008, 07:53 PMNonCommAlg
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.