Let R be a commutative ring with identity and let I be a two-generated ideal of R. We denote by ${\text{SL}}_2(R)$ the group of $2\times 2$ matrices over R with determinant 1. We study the action of ${\text{SL}}_2(R)$ by matrix right-multiplication on ${\mathrm{V}}_2(I)$, the set of generating pairs of I. Let ${\mathrm{Fitt}}_1(I)$ be the second Fitting ideal of I. Our main result asserts that ${\mathrm{V}}_2(I)/{\text{SL}}_2(R)$ identifies with a group of units of $R/{\mathrm{Fitt}}_1(I)$ 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 ${\text{SL}}_n(R)$ acts transitively on ${\mathrm{V}}_n(I)$ for every $n>2$. As an application, we derive a formula for the number of cusps of a modular group over a quadratic order.

令R为具有恒等式的交换环，令I为R的二生成理想。我们表示为${\text{SL}}_2(\mathrm{电阻})$ 一组 $2\times 2$行列式为 1 的R上的矩阵。我们研究了${\text{SL}}_2(\mathrm{电阻})$ 通过矩阵右乘 ${\mathrm{伏}}_2(\mathrm{一世})$，生成对的集合I。让${\mathrm{菲特}}_1(\mathrm{一世})$成为I的第二个拟合理想。我们的主要结果断言${\mathrm{伏}}_2(\mathrm{一世})/{\text{SL}}_2(\mathrm{电阻})$ 用一组单位标识 $\mathrm{电阻}/{\mathrm{菲特}}_1(\mathrm{一世})$如果I可以由两个正则元素生成，则通过行列式的自然推广。这个结果在几个低音环中得到了说明，我们还表明${\text{SL}}_n(\mathrm{电阻})$ 传递性地作用于 ${\mathrm{伏}}_n(\mathrm{一世})$ 对于每个 $n>2$. 作为一个应用，我们推导出一个关于二次阶模群的尖点数的公式。