| THE STRUCTURE OF ORTHOGONAL GROUPS OVER ARBITRARY COMMUTATIVE RINGS |
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| 作者姓名： | Li Fuan |
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| 作者单位： | Institute of |
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| 基金项目： | Projects supported by the Science Fund of the Chinese Academy of Sciences |
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| 摘 要： | Let R be an arbitrary commutative ring, and n an integer≥3. It is proved for any ideal J of R thatEO_(2n)(R, J)=EO_(2n)(R), EO_(2n)(J)]=EO_(2n)(R), EO_(2n)(R, J)]=EO_(2n)(R), O_(2n)(R,J)]=O_(2n)(R), EO_(2n)(R,J)].In particular, EO_(2n)(R, J) is a normal subgroupof O_(2n)(R). Furthermore, the problem of normal subgroups of O_(2n)(R) has an affirmative solution if and only if aR∩ Ann(2)=α~2 Ann(2) for each a in R. In particular, if 2 is not a zero divisor in R, then the problem of normal subgroups of O_(2n)(R) has an affirmative solution
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| 收稿时间： | 1987/1/24 0:00:00 |
THE STRUCTURE OF ORTHOGONAL GROUPS OVER ARBITRARY COMMUTATIVE RINGS |
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| Li Fuan.THE STRUCTURE OF ORTHOGONAL GROUPS OVER ARBITRARY COMMUTATIVE RINGS[J].Chinese Annals of Mathematics,Series B,1989,10(3):341-350. |
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| Authors: | Li Fuan |
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| Institution: | Department of Mattematics, Sichuan University, Chengdu, Sichuan, China. |
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| Abstract: | Let $R$ be an arbitrary commutative ring, and $n$ an integer $\ \ge 3\]$. It is proved for any ideal J of $R$ that
$$\\begin{array}{*{20}{c}}
{E{O_{2n}}(R,J) = E{O_{2n}}(R),E{O_{2n}}(J)] = E{O_{2n}}(R),E{O_{2n}}(R,J)]}\{ = E{O_{2n}}(R),{O_{2n}}(R,J)] = {O_{2n}}(R),E{O_{2n}}(R,J)]}
\end{array}\]$$
In particular, $\{E{O_{2n}}(R,J)}\]$ is a normal subgroup of $\{{O_{2n}}(R)}\]$. Furthermore, the problem of normal subgroups of $\{{O_{2n}}(R)}\]$ has an affirmative solution if and only if $\aB \cap Ann(2) = {a^2}Ann(2)\]$ for each $a$ in $R$. In particular, if 2 is not a zero divisor in R, then the problem of normal subgroups of $\{{O_{2n}}(R)}\]$ has an affirmative solution. |
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