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Math Help - units

  1. #1
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    units

    i read somewhere that a set of elements is a field iff all the elements are units except 0... but i thought that a ring R is a field iff they are commutative?

    from how i see, elements that are commutative might not be units.

    may i know what is wrong with my understanding?

    thanks
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  2. #2
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    Re: units

    A field is a commutative ring such that every nonzero element is a unit.

    \mathbb{R},\mathbb{Q},\mathbb{C} are all fields. That all three are commutative rings is a fact from elementary algebra. Also, in all three rings, we are able to divide by any nonzero element.
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  3. #3
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    Re: units

    that is almost correct. for a ring R, if every non-zero element is a unit, the proper term is division ring, as the multiplication may not be commutative.

    the standard example is the quaternions \mathbb{H} = \{a + bi + cj + dk: a,b,c,d \in \mathbb{R}\},

    where ij = k, jk = i, ki = j, ji = -k, kj = -i, ik = -j and i^2 = j^2 = k^2 = -1.

    however, it is a theorem (due to Wedderburn) that every finite division ring is indeed commutative: http://math.colgate.edu/math320/dlan...Wedderburn.pdf

    for an example of a finite ring that is NOT a field: consider \mathbb{Z}_6.

    note that (2)(3) = 0 (mod 6), so neither 2 nor 3 has an inverse (it would be like "dividing by 0").
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