Introduction. Lesieur and Croisot in [7] have generalized the classical primary decomposition theory for Noetherian modules over commutative rings to the tertiary decomposition theory for Noetherian modules over rings, which are not necessarily commutative, but which have a certain chain condition on ideals. Riley has shown in [8] that for finitely generated unitary modules over left Noetherian...

We study Serre’s condition (Sn) for tensor products of modules over a commutative noetherian local ring R. Specifically, we consider the following question. For finitely generated R-modules M and N, either which is (n+1)-Tor-rigid, if product M⊗RN satisfies (Sn+1), then does ToriR(M,N)=0 hold all i≥1? The aim this paper to give an affirmative answer question assume freeness on modules. As appli...

Let R be a commutative Noetherian local ring with residue field k. X resolving subcategory of finitely generated R-modules. This paper mainly studies when contains k or consists totally reflexive modules. It is proved that does so if closed under cosyzygies. A conjecture Dao and Takahashi also shown to hold in several cases.

We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are “tame” according to the Klingler–Levy analysis in [4], [5] and [6], in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases.

In this paper we study almost uniserial rings and modules. An R−module M is called almost uniserial if any two nonisomorphic submodules are linearly ordered by inclusion. A ring R is an almost left uniserial ring if R_R is almost uniserial. We give some necessary and sufficient condition for an Artinian ring to be almost left uniserial.

In this paper, we consider the Frobenius pushforward endofunctor F * ${F}_{\ast}$ of bounded derived category finitely generated modules over an $F$ -finite noetherian local ring. We completely determine categorical entropy in sense Dimitrov, Haiden, Katzarkov, and Kontsevich.

Let be a left and right noetherian ring and mod the category of finitely generated left -modules. In this article, we show the following results. 1 For a positive integer k, the condition that the subcategory of mod consisting of i-torsionfree modules coincides with the subcategory of mod consisting of i-syzygy modules for any 1 ≤ i ≤ k is left-right symmetric. 2 If is an -Gorenstein ring and N...

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck’s duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of ac...

In this paper we introduce techniques to gauge the torsion of the tensor product A ⊗R B of two finitely generated modules over a Noetherian ring R. The outlook is very different from the study of the rigidity of Tor carried out in the work of Auslander ([1]) and other authors. Here the emphasis in on the search for bounds for the torsion part of A⊗R B in terms of global invariants of A and of B...

We study the cancellation property of projective modules rank $2$ with a trivial determinant over Noetherian rings dimension $\leq 4$. If $R$ is smooth affine algebra $4$ an algebraically closed field $k$ such that $6 \in k^{\times}$, then we prove stably free $R$-modules are if and only Hermitian $K$-theory group $\tilde{V}_{SL} (R)$ trivial.