Injective dimension of Rees rings (Q1318867)

If \(A\) is a left and right noetherian ring, \(x\) is a central regular element of \(A\), and \(M\) is an \(A\)-module such that \(x\) is a nonzero divisor in \(M\), it is shown that the three injective dimensions \(\text{id}_ A(M)\), \(\text{id}_{A_ x}(M_ x)\) (localization) and \(\text{id}_{A/xA}(M/xM)\) are equal. Several applications are given for Rees rings.