# Hilbert–Burch theorem

${\displaystyle 0\rightarrow R^{m}{\stackrel {f}{\rightarrow }}R^{n}\rightarrow R\rightarrow R/I\rightarrow 0}$
is a free resolution of the R-module R/I, then m = n – 1 and the ideal I is aJ where a is a regular element of R and J, a depth-2 ideal, is the first Fitting ideal ${\displaystyle \mathrm {Fitt} _{1}I}$ of I, i.e., the ideal generated by the determinants of the minors of size m of the matrix of f.