# division ring

• Jun 2nd 2010, 05:48 AM
xixi
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 .
• Jun 2nd 2010, 06:34 PM
NonCommAlg
Quote:

Originally Posted by xixi
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.\$
• Jun 3rd 2010, 05:19 AM
xixi
Quote:

Originally Posted by NonCommAlg
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 ?