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Math Help - division ring

  1. #1
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    division ring

    Let R be a ring and Nil(R)=0 . Suppose that R=eRe \oplus fRf where e and f are idempotent elements and f=1-e . Let R_1=eRe and R_2=fRf . If R_1 \cong \mathbb{Z}_2 and |Max_l(R_2)|=1 then prove that R_2 is a division ring .
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  2. #2
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    Quote Originally Posted by xixi View Post
    Let R be a ring and Nil(R)=0 . Suppose that R=eRe \oplus fRf where e and f are idempotent elements and f=1-e . Let R_1=eRe and R_2=fRf . If R_1 \cong \mathbb{Z}_2 and |Max_l(R_2)|=1 then prove that R_2 is a division ring .
    first you need to clarify two thing for us:

    1) for noncommutative rings, there are more than one nilradical. so you need to give us the definition of Nil(R) in here.

    2) is |Max_l(R_2)|=1 supposed to mean that R_2 has only one maximal left ideral? (this is the first time i see this notation!)

    for now, just see that e,f are central, i.e. they are in the center of R and thus, since R_1 \cong \mathbb{Z}_2, we have R_1=\{0,e\}, \ R_2=fRf=Rf.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    first you need to clarify two thing for us:

    1) for noncommutative rings, there are more than one nilradical. so you need to give us the definition of Nil(R) in here.

    2) is |Max_l(R_2)|=1 supposed to mean that R_2 has only one maximal left ideral? (this is the first time i see this notation!)

    for now, just see that e,f are central, i.e. they are in the center of R and thus, since R_1 \cong \mathbb{Z}_2, we have R_1=\{0,e\}, \ R_2=fRf=Rf.
    1. By Nil(R)=0 I meant that R doesn't have any nonzero nilpotent element . Maybe I shouldn't have used this notation .
    2. Yes , |Max_l(R_2)|=1 means that R_2 has only one maximal left ideal .

    e,f are central and R_1=\{0,e\}, \ R_2=fRf=Rf. Now how do you prove that R_2 is a division ring ?
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