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Math Help - Rings with Unity

  1. #1
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    Rings with Unity

    I know that unity for a ring is just the multiplicative inverse for a ring, and that not every ring has a multiplicative inverse, so therefore not every ring has unity. I need to prove why a ring has at most one unity. I'm just not sure how to construct a well written proof for this. Any help would be greatly appreciated.
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by tfhawk View Post
    I know that unity for a ring is just the multiplicative inverse for a ring, and that not every ring has a multiplicative inverse, so therefore not every ring has unity. I need to prove why a ring has at most one unity. I'm just not sure how to construct a well written proof for this. Any help would be greatly appreciated.
    I'm not quite what you mean by multiplicative inverse for a ring?

    Unity is the multiplicative Identity

    PlanetMath: unity

    The idea of the proof is

    what if there are to elements of the Ring that are the multiplicative Identity call them e_1,e_2

    but then

    e_1 \cdot e_2 = e_2 becuase e_1 is an identity, and

    e_1 \cdot e_2 = e_1 becuase e_2

    This implies that

    e_1=e_1 \cdot e_2=e_2 \implies e_1=e_2

    So the identity is unique.
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  3. #3
    Super Member Gamma's Avatar
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    I think you are confusing the term "unit" and "unity" which is understandable. A unit is an element in a ring that has an inverse in the ring (multiplicative).
    unity is the identity element 1, such that 1r=r for all r in your ring. It is indeed uniques as \emptyset pointed out.

    It should also probably be noted that rings can certainly have unity without every element having an inverse. For instance \mathbb{Z} is a ring with unity 1, but only 1 and -1 have inverses.

    This is also a typical example to show that not every element in a ring is either a unit (invertible) or a zero divisor. This is an integral domain and has no zero divisors and only 2 units.
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