Approximations and the little finitistic dimension of Artinian rings (Q5956267)

Let \(R\) be an associative ring with identity element, let \(\mathcal S\) be a representative set of all simple right \(R\)-modules, let \(\mathcal P\) be the class of all finitely generated right \(R\)-modules having finite projective dimension, and let \(({\mathcal A},{\mathcal B})\) be the complete cotorsion theory cogenerated by \(\mathcal P\). The little finitistic dimension \(\text{fdim}(R)\) of \(R\) is defined as \(\sup\{\text{proj.dim}(M)\mid M\in{\mathcal P}\}\). For each \(S\in{\mathcal S}\), let \(X_S\to S\) be a special \(\mathcal A\)-precover of \(S\). The author proves that if \(R\) is right Artinian, then \(\text{fdim}(R)=\max\{\text{proj.dim}(X_S)\mid S\in{\mathcal S}\}\). As a corollary, one extends from Artin algebras to right Artinian rings the well-known Auslander-Reiten sufficient condition for the finiteness of the little finitistic dimension: if \(\mathcal P\) is contravariantly finite, then \(\text{fdim}(R)<\infty\).