Let \((R,{\mathfrak {m}},k)\) be a Noetherian local ring and let M be a finitely generated R-module. The main focus of this paper is to give positive answers to some long-standing homological conjectures over the idealization ring \(R < imes M.\) First, if N is a \(R < imes k\) -module, we show that the vanishing of \(\operatorname {Ext}_{R < imes k}^{i}(N,N\oplus (R < imes k))\) for some \(i\ge 3\) gives that N is free, and this provides a sharpened version of the Auslander–Reiten conjecture over \(R < imes k.\) Also, we give a characterization of the Betti numbers of an R-module over the idealization ring \(R < imes M\) and, as a biproduct, we derive that the Jorgensen–Leuschke conjecture holds for \(R < imes M.\) Further, we show that if Buchsbaum–Eisenbud–Horrocks and Total Rank conjectures over R holds, then holds for \(R < imes M.\) This establishes particular answers to both conjectures for modules with infinite projective dimension, especially when R is regular or a complete intersection ring. As applications of the idealization ring theory, we show that the Zariski–Lipman conjecture holds for any ring R provided the Betti numbers of the R-derivation module \(\operatorname {Der}_k(R),\) seen as \(R < imes k\) -module, satisfy the inequality \(\beta _{n}^{R < imes k}(\operatorname {Der}_k(R))\le \beta _{n-1}^{R < imes k}(\operatorname {Der}_k(R))\) for some \(n>0.\) Some implications regarding the Herzog–Vasconcelos conjecture are also provided.
