On the Countability of Noetherian Dimension of Modules

Abstract

It is shown that a module M has countable Noetherian dimension if and only if the lengths of ascending chains of submodules of M has a countable upper bound. This shows in particular that every submodule of a module with countable Noetherian dimension is countably generated. It is proved that modules with Noetherian dimension over locally Noetherian rings have countable Noetherian dimension. We also observe that ω^ω is a universal upper bound for the lengths of all chains in Artinian modules over commutative rings.     

Global dimension of Noetherian serial rings

The global dimension of Noetherian serial rings is studied. It is proved that if an indecomposable serial ring has infinite global dimension then it is Artinian and its quiver is a simple cycle. Using methods of the theory of right serial quivers, we give an upper estimate on the Loewy length of Artinian rings of finite global dimension. Applications to the calculation of the global dimension of tiled orders of width 2 are given.     

On noncommutative piecewise Noetherian rings

We extend the definition of a piecewise Noetherian ring to the noncommutative case, and investigate various properties of such rings. In particular, we show that a ring with Krull dimension is piecewise Noetherian. Certain fully bounded piecewise Noetherian rings have Gabriel dimension and exhibit the Gabriel correspondence between prime ideals and indecomposable injective modules.     

Global Theory of Lattice-Finite Noetherian Rings

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal. Presented by H. Tachikawa     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Noether模的Artin根

本文定义了Ｎｏｃｔｈｅｒ模的Ａｒｔｉｎ根，给出了Ｎｏｃｔｈｅｒ半局部环上有限生成模的Ａｒｔｉｎ根的刻划．最后，作为Ａｒｔｉｎ根理论的应用，给出了一个关于Ｃｏｈｅｎ－Ｍａｃａｕｌａｙ环的定理及一个模的用ｄｅｐｔｈ和ｋｒｕｌｌ维数表示的公式。     

Modules with Noetherian Spectrum

Let M be a module over a commutative ring R. A submodule P of M is called prime if P ≠ M and, whenever r ∈ R, e ∈ M, and re ∈ P, we have rM ? P or e ∈ P. We let Spec(M) denote the set of all prime submodules of M. Using a topology analogous to the Zariski topology for Spec(R), we establish necessary and sufficient conditions for Spec(M) to be a Noetherian space. We produce some examples of modules with Noetherian spectrum that have not appeared in the literature previously. In particular, Laskerian modules and faithfully flat modules over Laskerian rings have Noetherian spectra. (The term Laskerian is defined in Section 3.)     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

Jacobson's conjecture and modules over fully bounded Noetherian rings

The object of this paper is to investigate finitely generated modules and injective modules over fully bounded Noetherian rings. Our main results on f.g. modules (Theorems 3.1 and 3.4) provide an analog of the Jordan-Holder Theorem and seem to be new even for commutative Noetherian rings. They imply the validity of Jacobson's conjecture for fully bounded Noetherian rings. The main result on injectives (Theorem 5.3) describes f.g. submodules of an indecomposible injective and shows how it can be built from indecomposible injectives over certain Artinian rings.     

Noetherian Rings whose Modules are Prime Serial

A theorem due to Nakayama and Skornyakov states that “a ring R is an Artinian serial ring if and only if all left R-modules are serial” and a theorem due to Warfield state that “a Noetherian ring R is serial if and only if every finitely generated left R-module is serial”. We say that an R-module M is prime uniserial (?-uniserial, for short) if for every pair P, Q of prime submodules of M either \(P\subseteq Q\) or \(Q\subseteq P\), and we say that M is prime serial (?-serial, for short) if it is a direct sum of ?-uniserial modules. Therefore, two interesting natural questions of this sort are: “Which rings have the property that every module is ?-serial?” and “Which rings have the property that every finitely generated module is ?-serial?” Most recently, in our paper, Prime uniserial modules and rings (submitted), we considered these questions in the context of commutative rings. The goal of this paper is to answer these questions in the case R is a Noetherian ring in which all idempotents are central or R is a left Artinian ring.     

Uniserial Noetherian centrally essential rings

Abstract

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings.     

Homological Properties of Fully Bounded Noetherian Rings

Let R be a fully bounded Noetherian ring of finite global dimension.Then we prove that K dim (R) gldim (R). If, in addition, Ris local, in the sense that R/J(R) is simple Artinian, thenwe prove that R is Auslander-regular and satisfies a versionof the Cohen–Macaulay property. As a consequence, we showthat a local fully bounded Noetherian ring of finite globaldimension is isomorphic to a matrix ring over a local domain,and a maximal order in its simple Artinian quotient ring.     

On the small Krull dimension of modules

We introduce and study the concept of small Krull dimension of a module which is Krull-like dimension extension of the concept of DCC on small submodules. Using this concept we extend some of the basic results for modules with this dimension, which are almost similar to the basic properties of modules with Krull dimension. When for a module A with small Krull dimension, whose Rad(A) is quotient finite dimensional, then these two dimensions for Rad(A) coincide. In particular, we prove that if an R-module A has finite hollow dimension, then A has small Krull dimension if and only if it has Krull dimension. Consequently, we show that if A has properties AB5* and qfd, then A has s.Krull dimension if and only if A has Krull dimension.     

Completion of Modules over Serial, Right Noetherian Rings

Linear topology defined on an arbitrary right module over a right Noetherian serial ring R enables one to describe the reduced, pure injective R-modules as modules that are complete in this topology. With the use of the completion of modules, the pure injective envelope of any right R-module is constructed. Bibliography: 8 titles.     

Local Duality for Modules over Noetherian Commutative Rings

Some applications of the general theorem on the existence of local duality for modules over Noetherian commutative rings are given. Let be a Noetherian commutative ring, let be a set of maximal ideals in , and let . Then the category of Artinian modules is dual to the category of Noetherian modules. Several structural results are proved, including the theorem on the structure of Artinian modules over principal ideal domains. For rings of special kinds, double centralizer theorems are proved. Bibliography: 5 titles.     

The Krull dimension of the module category over right noetherian serial rings

For a right Noetherian serial ring R that is not Artinian, it is proved that the Krull dimension of the category of finitely generated right R-modules is equal to one. Bibliography: 17titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 73–86.     

Modules with Chain Conditions on Non-essential Submodules

Abstract

We investigate when modules which satisfy the descending (respectively, ascending) chain condition on non-essential submodules are uniform or Artinian (respectively, Noetherian).     

On the artinianness of the local homology modules

In [9] a local homology theory for Artinian modules over commutative rings, which is dual to the local cohomology theory for Noetherian modules, introduced and in [1] the main result of [9] extended. In this note we prove that the local homology modules of an Artinian module over a commutative ring (with identity) are Artinian.     

Basic Submodules of Modules Over Serial, Right Noetherian Rings. II

In the preceding paper, the authors published an existence theorem for basic submodules of right modules over right Noetherian, serial rings. The aim of the present paper is to prove a uniqueness theorem for basic submodules over such rings. Bibliography: 13 titles.