Equivalent generating pairs of an ideal of a commutative ring

Abstract

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.