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Thread: 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 $\displaystyle e_1,e_2$

    but then

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

    $\displaystyle e_1 \cdot e_2 = e_1$ becuase $\displaystyle e_2$

    This implies that

    $\displaystyle e_1=e_1 \cdot e_2=e_2 \implies e_1=e_2$

    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 $\displaystyle \emptyset$ pointed out.

    It should also probably be noted that rings can certainly have unity without every element having an inverse. For instance $\displaystyle \mathbb{Z}$ 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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