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.

We show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[M]$, is weakly co-semisimple if and only if it is regular in $si[M]$. As a consequence, we observe that every semi-artinian ring is regular in the sense of von Neumann if and only if its simple modules are $FP$-injective.

In this paper, we introduce a generalization of Hilbert $C^*$-modules which are pre-Finsler modules, namely, $C^{*}$-semi-inner product spaces. Some properties and results of such spaces are investigated, specially the orthogonality in these spaces will be considered. We then study bounded linear operators on $C^{*}$-semi-inner product spaces.

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.

In this paper we look at the properties of modules and prime ideals in finite dimensional noetherian rings. This paper is divided into four sections. The first section deals with noetherian one-dimensional rings. Section Two deals with what we define a “zero minimum rings” and explores necessary and sufficient conditions for the property to hold. In Section Three, we come to the minimal prime i...

The partial functions under disjoint-domain sums and functional composition do not form a field, and thus conventional linear algebra is not applicable. However they can be regarded as a so-ring, an algebraic structure possessing a natural partial ordering, an infinitary partial addition and a binary multiplication, subject to a set of axioms. In this paper, we introduce the notions of 2-absorb...