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Thread: division ring

  1. #1
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    division ring

    Let $\displaystyle R$ be a ring and $\displaystyle Nil(R)=0$ . Suppose that $\displaystyle R=eRe \oplus fRf$ where $\displaystyle e$ and $\displaystyle f$ are idempotent elements and $\displaystyle f=1-e$ . Let $\displaystyle R_1=eRe$ and $\displaystyle R_2=fRf$ . If $\displaystyle R_1 \cong \mathbb{Z}_2$ and $\displaystyle |Max_l(R_2)|=1$ then prove that $\displaystyle R_2$ is a division ring .
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  2. #2
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    Quote Originally Posted by xixi View Post
    Let $\displaystyle R$ be a ring and $\displaystyle Nil(R)=0$ . Suppose that $\displaystyle R=eRe \oplus fRf$ where $\displaystyle e$ and $\displaystyle f$ are idempotent elements and $\displaystyle f=1-e$ . Let $\displaystyle R_1=eRe$ and $\displaystyle R_2=fRf$ . If $\displaystyle R_1 \cong \mathbb{Z}_2$ and $\displaystyle |Max_l(R_2)|=1$ then prove that $\displaystyle R_2$ 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 $\displaystyle Nil(R)$ in here.

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

    for now, just see that $\displaystyle e,f$ are central, i.e. they are in the center of $\displaystyle R$ and thus, since $\displaystyle R_1 \cong \mathbb{Z}_2,$ we have $\displaystyle R_1=\{0,e\}, \ R_2=fRf=Rf.$
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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 $\displaystyle Nil(R)$ in here.

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

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

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