论文标题
等效产生的成对的理想是通勤环的理想
Equivalent generating pairs of an ideal of a commutative ring
论文作者
论文摘要
让$ r $成为具有身份的交换戒指，让$ i $是$ r $的两生的理想。我们用$ \ operatorname {sl} _2（r）$ $ 2 \ times 2 $矩阵$ r $，$ 1 $ 1 $。我们研究了$ \ operatorname {sl} _2（r）$的动作，由矩阵right-multiplication on $ \ permatatorname {v} _2（i）$，$ i $的生成对集。令$ \ operatorname {fitt} _1（i）$为$ i $的第二个拟合理想。我们的主要结果断言，$ \ operatorName {v} _2（i）/\ propatatorName {sl} _2（r）$识别一组$ r/\ operatatorName {fitt} _1（fitt} _1（i）$的单位，如果$ $ $ $ $可以生成两份常规元素。该结果在几个低音环中进行了说明，我们还表明，每$ n> 2 $，$ \ operatorname {sl} _n（r）$在$ \ operatatorName {v} _n（i）$上进行透支。作为一个应用程序，我们在二次顺序上得出一个模块组的尖数的公式。
Let $R$ be a commutative ring with identity and let $I$ be a two-generated ideal of $R$. We denote by $\operatorname{SL}_2(R)$ the group of $2 \times 2$ matrices over $R$ with determinant $1$. We study the action of $\operatorname{SL}_2(R)$ by matrix right-multiplication on $\operatorname{V}_2(I)$, the set of generating pairs of $I$. Let $\operatorname{Fitt}_1(I)$ be the second Fitting ideal of $I$. Our main result asserts that $\operatorname{V}_2(I)/\operatorname{SL}_2(R)$ identifies with a group of units of $R/\operatorname{Fitt}_1(I)$ via a natural generalization of the determinant if $I$ can be generated by two regular elements. This result is illustrated in several Bass rings for which we also show that $\operatorname{SL}_n(R)$ acts transitively on $\operatorname{V}_n(I)$ for every $n > 2$. As an application, we derive a formula for the number of cusps of a modular group over a quadratic order.
