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Math Help - Prove it is a field

  1. #1
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    Prove 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

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  2. #2
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    Quote Originally Posted by mandy123 View Post
    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 ax=b has a unique solution for any a \neq 0. if F is a field and 0 \neq a \in F, then from ax=b you'll get x=a^{-1}b and you're done.

    but the converse is less trivial: so we have that any equation ax=b, \ a \neq 0, has a unique solution in F. first we show that F has no zero divisor: suppose that uv=0 and u \neq 0. now the

    equation ux=0 has two solutions x=0,v. thus v=0. this proves that F has no zero divisor. next we show that F is unitary: choose any 0 \neq a \in F. then ax_0=a for some x_0 \in F. now

    let 0 \neq b \in F. then there exists x_1 \in F such that bx_1=b. but then ab(x_1-x_0)=0. since F has no zero divisor, we must have x_1=x_0. clearly 0x_0=0. thus: \forall b \in F: bx_0=b. therefore

    x_0=1_F. finally for any 0 \neq a \in F the equation ax=1 has a (unique) solution, which means any non-zero element of F is invertible. \Box
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