Let \(R\) be a ring. A sequence of \(R\)-modules and \(R\)-linear maps \[ M_0 \mapsvia{f_1} M_1 \mapsvia{f_2} \cdots \mapsvia{f_n} M_n \] is exact at \(M_i\), for \(0 < i < n\), if \(\im f_i = \ker f_{i+1}\); it is exact if it is exact at \(M_i\) for every \(0 < i < n\).
A sequence of \(R\)-module maps \[ 0 \mapsvia{d_1} A \mapsvia{d_2} B \mapsvia{d_3} C \mapsvia{d_4} 0 \] is exact if and only if \(\im d_i = \ker d_{i+1}\) for \(i = 1, 2, 3\), that is, if and only if \(d_2\) is injective, \(\im d_2 = \ker d_3\), and \(d_3\) is surjective.