Dimension on non-essential submodules

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an [Formula: see text]-module, where [Formula: see text] is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring [Formula: see text] with non-essential Noetherian dimension is uniform.

Noetherian Dimension and Co-localization of Artinian Modules over Local Rings

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dim RA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dim RA = dim (R/ Ann RA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.

On $\alpha$-Short Modules

We introduce and study the concept of $\alpha$-short modules (a $0$-short module is just a short module, i.e., for each submodule $N$ of a module $M$, either $N$ or $\frac{M}{N}$ is Noetherian). Using this concept we extend some of the basic results of short modules to $\alpha$-short modules. In particular, we show that if $M$ is an $\alpha$-short module, where $\alpha$ is a countable ordinal, then every submodule of $M$ is countably generated. We observe that if $M$ is an $\alpha$-short module then the Noetherian dimension of $M$ is either $\alpha$ or $\alpha+1$. In particular, if $R$ is a semiprime ring, then $R$ is $\alpha$-short as an $R$-module if and only if its Noetherian dimension is $\alpha$.

ON G-TYPE DOMAINS

In this paper, we introduce and study the notion of G-type domains (a domain R is G-type if its quotient field is countably generated R-algebra). We extend some of the basic properties of G-domains to G-type domains. It's observed that a prime ideal of R[x1, x2,…,xn,…] is G-type if and only if its contractions in R, R[x1, x2,…,xn] for all n ≥ 1 are G-type. Using this concept we give a natural proof of the well-known Hilbert Nullstellensatz in infinite countable-dimensional spaces. Characterizations of Noetherian G-type domains, Noetherian G-type domains with the countable prime avoidance property are given. As a consequence, we observe that in complete Noetherian semi-local rings, G-type ideals and G-ideals are the same. Rings with countable Noetherian dimension which are direct sum of G-type domains are fully determined. Finally, we characterize Noetherian rings in which G-type ideals are maximal.

Local Cohomology and Sequentially Cohen-Macaulay Modules

Let (R, 𝔪) be a commutative Noetherian local ring and N a finitely generated R-module with dim M =d. It is shown that M is a sequentially Cohen-Macaulay module if and only if the modules [Formula: see text] are either 0 or co-Cohen-Macaulay of Noetherian dimension i for all 0 ≤ i ≤ d.

Matlis Duality and Finiteness Properties of Generalized Local Cohomology Modules

Let [Formula: see text] be an ideal of a commutative Noetherian local ring [Formula: see text] and M, N be two finitely generated R-modules such that M is of finite projective dimension n. Let t be a positive integer. We show that if there exists a regular sequence [Formula: see text] with [Formula: see text] and the i-th local cohomology module [Formula: see text] of N with respect to [Formula: see text] is zero for all i > t, then [Formula: see text], where D(-):= Hom R(-,E). Also, we prove that if N is a Cohen-Macaulay R-module of dimension d, then the generalized local cohomology module [Formula: see text] is co-Cohen-Macaulay of Noetherian dimension d. Finally, with an elementary proof, we show that [Formula: see text] is finite.

Local Homology, Cohen–Macaulayness and Cohen–Macaulayfications

Let (R,𝔪) be a local ring, X an artinian R-module of noetherian dimension d; let x1,…,xd ∈ 𝔪 be such that 0:X (x1,…,xd)R has finite length. We show by an example that [Formula: see text] is not finite as an R-module in general; it is finite if we assume R is complete. This answers a question posed by Tang. As a first application of the latter finiteness result, we give a necessary condition for a finite module to be Cohen–Macaulay; secondly we propose a notion of Cohen–Macaulayfication and prove its uniqueness; finally we show that this new notion of Cohen–Macaulayfication is a direct generalization of a notion of Cohen–Macaulayfication introduced by Goto.

Morley degree in unidimensional compact complex spaces

Let be the category of all reduced compact complex spaces, viewed as a multi-sorted first order structure, in the standard way. Let be a sub-category of . which is closed under the taking of products and analytic subsets, and whose morphisms include the projections. Under the assumption that Th() is unidimensional. we show that Morley rank is equal to Noetherian dimension, in any elementary extension of . As a result, we are able to show that Morley degree is definable in Th(). when Th() is unidimensional.