Ring homomorphisms and local rings with quasi-decomposable maximal ideal (Q6600710)

\textit{T. Ogoma} [J. Math. Kyoto Univ. 24, 27--48 (1984; Zbl 0593.13011)] proved that the class of fiber product rings coincides with the class of local rings with decomposable maximal ideal. [\textit{S. Nasseh} and \textit{R. Takahashi} [Math. Proc. Camb. Philos. Soc. 168, No. 2, 305--322 (2020; Zbl 1433.13017)] introduced the class of local rings with quasi-decomposable maximal ideals, i.e., deform to fiber product rings. Note that fiber product rings have depth at most \(1\), hence, generally, non-Cohen-Macaulay, but Nasseh and Takahashi provided many examples of Cohen-Macaulay local rings with quasi-decomposable maximal ideals. In the first result of the paper, the authors provide examples of local rings with quasi-decomposable maximal ideals coming from edge ideals of graphs.\N\NOn another aspect, local rings with quasi-decomposable maximal ideals are Tor-friendly, i.e., for every pair \((M, N)\) of finitely generated \(R\)-modules, the condition \(\operatorname{Tor}_i^R(M, N)\) for \(i \gg 0\) implies that either \(M\) or \(N\) has finite projective dimension. The authors then studied the Tor-friendly properties of rings that are deformations of local rings with quasi-decomposable maximal ideals.