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 ${\mathrm{SL}}_2(R)$ the group of $2\times 2$ matrices over $R$ with determinant $1$. We study the action of ${\mathrm{SL}}_2(R)$ by matrix right-multiplication on $V_2(I)$, the set of generating pairs of $I$. Let ${\mathrm{Fitt}}_1(I)$ be the second Fitting ideal of $I$. Our main result asserts that $V_2(I)/{\mathrm{SL}}_2(R)$ 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 ${\mathrm{SL}}_n(R)$ acts transitively on $V_n(I)$ for every $n>2$. As an application, we derive a formula for the number of cusps of a modular group over a quadratic order.

26 pages, no figure. Minor changes: fix a couple of embarrassing typos