scholarly journals Serial rings with krull dimension

1990 ◽  
Vol 32 (1) ◽  
pp. 71-78 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Structure Theorem ◽  
Krull Dimension ◽  
Serial Ring ◽  
Direct Sum

A module is said to be serial if it has a unique chain of submodules, and a ring is serial if it is a direct sum of serial right ideals and a direct sum of serial left ideals. The serial rings of Krull dimension 0 are the Artinian serial (or generalised uniserial) rings studied by Nakayama and for which there is an extensive theory (see for example [4]). Warfield in [10] extended the theory to the non-Artinian case. In particular he showed that a Noetherian serial ring is a direct sum of Artinian serial rings and prime Noetherian serial rings, and he gave a structure theorem in the prime Noetherian case. A Noetherian non-Artinian serial ring has Krull dimension 1. Serial rings of arbitrary Krull dimension have been studied by Wright ([9], [12], [13], [14]) with special results being proved when the Krull dimension is 1 or 2.

scholarly journals Injective Modules Over Non-Artinian Serial Rings

1988 ◽  
Vol 44 (2) ◽  
pp. 242-251 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Affirmative Answer ◽  
Primitive Idempotent ◽  
Injective Hull ◽  
Finitely Generated ◽  
Serial Ring ◽  
Direct Sum ◽  
Injective Modules

AbstractA module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

The structure of unique factorization rings

1970 ◽  
Vol 67 (3) ◽  
pp. 535-540 ◽  
Author(s):  
C. R. Fletcher
Keyword(s):  
Finite Number ◽  
Principal Ideal ◽  
Structure Theorem ◽  
Unique Factorization ◽  
Direct Sum ◽  
Unique Factorization Domains

1. Introduction. In (1) we proved that the direct sum of a finite number of unique factorization rings is a unique factorization ring (UFR), and in particular that the direct sum of a finite number of unique factorization domains (UFD's) is a UFR. The converse, however, does not hold i.e. not every UFR can be expressed as a direct sum of UFD's. Here we investigate the structure of UFR's and show that every UFR is a finite direct sum of UFD's and of special UFR's. There is thus a relationship with the structure theorem for principal ideal rings ((2), p. 245).

scholarly journals Classes of modules with many direct summands

1995 ◽  
Vol 59 (1) ◽  
pp. 8-19 ◽  
Author(s):  
I. Al-Khazzi ◽  
P. F. Smith
Keyword(s):  
Direct Summand ◽  
Krull Dimension ◽  
Direct Sum ◽  
Semisimple Module

AbstractLet R be any ring with identity, M a unital right R-module and α ≥ 0 an ordinal. Then M is a direct sum of a semisimple module and a module having Krull dimension at most α if and only if for every submodule N of M there exists a direct summand K of M such that K ⊆ N and N/K has Krull dimension at most α.

A CONSTRUCTION OF RIGHT ARTINIAN RIGHT SERIAL RINGS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350057
Author(s):  
SURJEET SINGH
Keyword(s):  
Homomorphic Image ◽  
Artinian Ring ◽  
Finitely Generated ◽  
Serial Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Non Local ◽  
Universal Construction ◽  
Generating System

A ring R is said to be right serial, if it is a direct sum of right ideals which are uniserial. A ring that is right serial need not be left serial. Right artinian, right serial ring naturally arise in the study of artinian rings satisfying certain conditions. For example, if an artinian ring R is such that all finitely generated indecomposable right R-modules are uniform or all finitely generated indecomposable left R-modules are local, then R is right serial. Such rings have been studied by many authors including Ivanov, Singh and Bleehed, and Tachikawa. In this paper, a universal construction of a class of indecomposable, non-local, basic, right artinian, right serial rings is given. The construction depends on a right artinian, right serial ring generating system X, which gives rise to a tensor ring T(L). It is proved that any basic right artinian, right serial ring is a homomorphic image of one such T(L).

On commutative rings whose ideals are direct sums of cyclic modules

2019 ◽  
Vol 18 (02) ◽  
pp. 1950039
Author(s):  
A. Ghorbani ◽  
M. Naji-Esfahani
Keyword(s):  
Maximal Ideal ◽  
Structure Theorem ◽  
Commutative Rings ◽  
Local Dimension ◽  
Direct Sums ◽  
Principal Ideal Ring ◽  
Direct Sum ◽  
Dimension Formula ◽  
Local Case ◽  
The Family

A generalization of Köthe rings is the family of rings whose ideals are direct sums of cyclic modules. These rings were previously studied in the commutative local case. This motivated us to study commutative rings with local dimension whose ideals are direct sums of cyclic modules. First, we obtain an structure theorem for rings of local dimension [Formula: see text]. Then we conclude that, for a ring of local dimension [Formula: see text], every ideal of [Formula: see text] is a direct sum of cyclic modules if and only if every indecomposable ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if every maximal ideal of [Formula: see text] is cyclic if and only if [Formula: see text] is a principal ideal ring.

scholarly journals Annihilator and complemented banach*-algebras

1972 ◽  
Vol 13 (1) ◽  
pp. 47-66 ◽  
Author(s):  
B. J. Tomiuk ◽  
Pak-Ken Wong
Keyword(s):  
Banach Algebra ◽  
Structure Theorem ◽  
Banach Algebras ◽  
Direct Sum ◽  
Dense Subalgebra

The study of complemented Banach*-algebras taken up in [1] was confined mainly toB*-algebras. In the present paper we extend this study to (right) complemented Banach*-algebras in whichx*x= 0 impliesx= 0. We show that ifAis such an algebra then every closed two-sided ideal ofAis a *-ideal. Using this fact we obtain a structure theorem forAwhich states that ifAis semi-simple thenAcan be expressed as a topological direct sum of minimal closed two sided ideals each of which is a complemented Banach*-algebra. It follows thatAis an A*-algebra and is a dense subalgebra of a dual B*-algebraU, which is determined uniquely up to *-isomorphism.

Rings all of whose prime serial modules are serial

2017 ◽  
Vol 16 (10) ◽  
pp. 1750193
Author(s):  
M. Behboodi ◽  
Z. Fazelpour ◽  
M. R. Vedadi
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Invariant Property ◽  
Serial Ring ◽  
Direct Sum ◽  
Morita Equivalent ◽  
Morita Invariant ◽  
Prime Submodules ◽  
Uniserial Modules

It is well known that the concept of left serial ring is a Morita invariant property and a theorem due to Nakayama and Skornyakov states that “for a ring [Formula: see text], all left [Formula: see text]-modules are serial if and only if [Formula: see text] is an Artinian serial ring”. Most recently the notions of “prime uniserial modules” and “prime serial modules” have been introduced and studied by Behboodi and Fazelpour in [Prime uniserial modules and rings, submitted; Noetherian rings whose modules are prime serial, Algebras and Represent. Theory 19(4) (2016) 11 pp]. An [Formula: see text]-module [Formula: see text] is called prime uniserial ( [Formula: see text]-uniserial) if its prime submodules are linearly ordered with respect to inclusion, and an [Formula: see text]-module [Formula: see text] is called prime serial ( [Formula: see text]-serial) if [Formula: see text] is a direct sum of [Formula: see text]-uniserial modules. In this paper, it is shown that the [Formula: see text]-serial property is a Morita invariant property. Also, we study what happens if, in the above Nakayama–Skornyakov Theorem, instead of considering rings for which all modules are serial, we consider rings for which every [Formula: see text]-serial module is serial. Let [Formula: see text] be Morita equivalent to a commutative ring [Formula: see text]. It is shown that every [Formula: see text]-uniserial left [Formula: see text]-module is uniserial if and only if [Formula: see text] is a zero-dimensional arithmetic ring with [Formula: see text] T-nilpotent. Moreover, if [Formula: see text] is Noetherian, then every [Formula: see text]-serial left [Formula: see text]-module is serial if and only if [Formula: see text] is serial ring with dim[Formula: see text].

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

On Lattice-Ordered Rees Matrix Γ-Semigroups

2013 ◽  
Vol 59 (1) ◽  
pp. 209-218 ◽  
Author(s):  
Kostaq Hila ◽  
Edmond Pisha
Keyword(s):  
Structure Theorem ◽  
Lattice Of Congruences

Abstract The purpose of this paper is to introduce and give some properties of l-Rees matrix Γ-semigroups. Generalizing the results given by Guowei and Ping, concerning the congruences and lattice of congruences on regular Rees matrix Γ-semigroups, the structure theorem of l-congruences lattice of l - Γ-semigroup M = μº(G : I; L; Γe) is given, from which it follows that this l-congruences lattice is distributive.

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