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Math Help - ring and field

  1. #1
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    ring and field

    how do we directly prove that if a ring is a field if and only if (0) is a maximal ideal? but without using the theorem that I is maximal if and only if R/I is a field

    can anyone help me please ??
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    how do we directly prove that if a ring is a field if and only if (0) is a maximal ideal? but without using the theorem that I is maximal if and only if R/I is a field

    can anyone help me please ??

    A unitary ring R is a field iff all its non-zero elements are invertible iff the principal ideal

    \langle r\rangle\,,\forall\,0\neq r\in R is the whole ring iff its (unique) proper ideal {0} is maximal.

    Tonio
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