We introduce a new notion of commutative noetherian local rings, which we call dominant.
We explore fundamental properties of dominant local rings and compare them with other …

Let (R, 𝔪) be a commutative noetherian local ring. In this paper, we prove that if 𝔪 is
decomposable, then for any finitely generated R-module M of infinite projective dimension 𝔪 …

Consider a non-trivial fiber product $ R= S\times _kT $ of local rings $ S $, $ T $ with
common residue field $ k $. Given two finitely generated $ R $-modules $ M $ and $ N $, we …

We construct a local Cohen–Macaulay ring R with a prime ideal p∈ Spec (R) such that R
satisfies the uniform Auslander condition (UAC), but the localization R p does not satisfy …

We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can
be used as test modules, which detect finite homological dimensions of modules. We prove …

Conditions on the Koszul complex of a noetherian local ring R guarantee that Tor i R (M, N)
is nonzero for infinitely many i, when M and N are finitely generated R-modules of infinite …

Let n be a positive integer, and let A be a strongly commutative differential graded (DG)
algebra over a commutative ring R. Assume that B= A [X 1,…, X n] is a polynomial extension …

The notion of naive lifting of DG modules was introduced by the authors in [16, 17] for the
purpose of studying problems in homological commutative algebra that involve self …

The Burch index is a new invariant of a local ring R whose positivity implies a kind of
linearity in resolutions of R-modules. We show that if R has depth zero and Burch index at …

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by
Nasseh and Takahashi. In separate works, the authors of the present paper showed that …