Dimension on non-essential submodules

2019 ◽  
Vol 18 (05) ◽  
pp. 1950089 ◽  
Author(s):  
Maryam Davoudian
Keyword(s):  
Associative Ring ◽  
Semiprime Ring ◽  
Krull Dimension ◽  
Goldie Dimension ◽  
Ordinal Numbers ◽  
Noetherian Dimension

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.

Stable Rings

1980 ◽  
Vol 23 (2) ◽  
pp. 173-178 ◽  
Author(s):  
S. S. Page
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Commutative Rings ◽  
Von Neumann ◽  
Goldie Dimension ◽  
Von Neumann Regular

Let R be an associative ring with identity. If R is von- Neumann regular of a left v-ring, then for each left ideal, I, we have I2 = I. In this note we study rings such that for each left ideal L there exists an integer n = n(L)>0 such that Ln = Ln+1. We call such rings stable rings. We completely describe the stable commutative rings. These descriptions give rise to concepts related to, but more general than, finite Goldie dimension and T-nilpotence, and a notion of power pure.

scholarly journals GOLDIE DIMENSION, DUAL KRULL DIMENSION AND SUBDIRECT IRREDUCIBILITY

2010 ◽  
Vol 52 (A) ◽  
pp. 19-32 ◽  
Author(s):  
TOMA ALBU
Keyword(s):  
Krull Dimension ◽  
Subdirectly Irreducible ◽  
Modular Lattices ◽  
Survey Paper ◽  
Goldie Dimension ◽  
Subdirect Irreducibility ◽  
Upper Continuous ◽  
Torsion Theories ◽  
Quotient Modules ◽  
Noetherian Modules

AbstractIn this survey paper we present some results relating the Goldie dimension, dual Krull dimension and subdirect irreducibility in modules, torsion theories, Grothendieck categories and lattices. Our interest in studying this topic is rooted in a nice module theoretical result of Carl Faith [Commun. Algebra27 (1999), 1807–1810], characterizing Noetherian modules M by means of the finiteness of the Goldie dimension of all its quotient modules and the ACC on its subdirectly irreducible submodules. Thus, we extend his result in a dual Krull dimension setting and consider its dualization, not only in modules, but also in upper continuous modular lattices, with applications to torsion theories and Grothendieck categories.

scholarly journals Goldie dimensions of quotient modules

2001 ◽  
Vol 71 (1) ◽  
pp. 11-19
Author(s):  
John Dauns
Keyword(s):  
Associative Ring ◽  
Infinite Cardinal ◽  
Goldie Dimension ◽  
Injective Modules ◽  
Subdirect Products ◽  
Quotient Modules ◽  
Quotient Property

AbstractFor an infinite cardinal ℵ an associative ring R is quotient ℵ<-dimensional if the generalized Goldie dimension of all right quotient modules of RR are strictly less than ℵ. This latter quotient property of RR is characterized in terms of certain essential submodules of cyclic modules being generated by less than ℵ elements, and also in terms of weak injectivity and tightness properties of certain subdirect products of injective modules. The above is the higher cardinal analogue of the known theory in the finite ℵ = ℵ0 case.

scholarly journals Modules Whose Cyclic Submodules Have Finite Dimension

1976 ◽  
Vol 19 (1) ◽  
pp. 1-6 ◽  
Author(s):  
David Berry
Keyword(s):  
Associative Ring ◽  
Finite Dimension ◽  
Essential Extension ◽  
Injective Hull ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Cyclic Submodules ◽  
Unitary Right

R denotes an associative ring with identity. Module means unitary right R-module. A module has finite Goldie dimension over R if it does not contain an infinite direct sum of nonzero submodules [6]. We say R has finite (right) dimension if it has finite dimension as a right R-module. We denote the fact that M has finite dimension by dim (M)<∞.A nonzero submodule N of a module M is large in M if N has nontrivial intersection with nonzero submodules of M [7]. In this case M is called an essential extension of N. N⊆′M will denote N is essential (large) in M. If N has no proper essential extension in M, then N is closed in M. An injective essential extension of M, denoted I(M), is called the injective hull of M.

scholarly journals Higher commutators, ideals and cardinality

1996 ◽  
Vol 54 (1) ◽  
pp. 41-54 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Finite Index ◽  
Associative Ring ◽  
Semiprime Ring ◽  
Torsion Free ◽  
The Ideal

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [[R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2cardT and there is a central ideal of R of index at most 2cardT in R.

On $\alpha$-Short Modules

2014 ◽  
Vol 114 (1) ◽  
pp. 26 ◽  
Author(s):  
M. Davoudian ◽  
O. A. S. Karamzadeh ◽  
N. Shirali
Keyword(s):  
Semiprime Ring ◽  
Countable Ordinal ◽  
Noetherian Dimension

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 $\alpha$.

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

2009 ◽  
Vol 37 (2) ◽  
pp. 650-662 ◽  
Author(s):  
J. Hashemi ◽  
O. A. S. Karamzadeh ◽  
N. Shirali
Keyword(s):  
Krull Dimension ◽  
Noetherian Dimension

On rings and algebras with derivations

2016 ◽  
Vol 15 (06) ◽  
pp. 1650107 ◽  
Author(s):  
Shakir Ali ◽  
Mohammad Salahuddin Khan ◽  
Abdul Nadim Khan ◽  
Najat M. Muthana
Keyword(s):  
Banach Algebra ◽  
Associative Ring ◽  
Semiprime Ring ◽  
Derivation Formula ◽  
Positive Integers ◽  
Rings And Algebras

Let [Formula: see text] be an associative ring with center [Formula: see text] The objective of this paper is to discuss the commutativity of a semiprime ring [Formula: see text] which admits a derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text] where [Formula: see text] and [Formula: see text] are fixed positive integers. Finally, we apply these purely ring theoretic results to obtain commutativity of Banach algebra via derivation.

scholarly journals A characterization of left semiartinian rings

1974 ◽  
Vol 11 (3) ◽  
pp. 425-428 ◽  
Author(s):  
Jonathan S. Golan
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Krull Dimension ◽  
Semiartinian Ring ◽  
Torsion Theories ◽  
Necessary And Sufficient

In defining the torsion-theoretic Krull dimension of an associative ring R we make use of a function δ from the complete lattice of all subsets of the torsion-theoretic spectrum of R to the complete lattice of all hereditary torsion theories on R-mod. In this note we give necessary and sufficient conditions for δ to be injective, surjective, and bijective. In particular, δ is bijective if and only if R is a left semiartinian ring.

A study of derivable mappings of semiprime rings

2018 ◽  
Vol 7 (1-2) ◽  
pp. 19-26
Author(s):  
Gurninder S. Sandhu ◽  
Deepak Kumara
Keyword(s):  
Associative Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Commutativity Theorems ◽  
Semiprime Rings

Throughout this note, \(R\) denotes an associative ring and \(C(R)\) be the center of \(R\). In this paper, it isproved that a non-central Lie ideal \(L\) of a semiprime ring \(R\) contains a nonzero ideal of \(R\) and this result isused to obtain several commutativity theorems of \(R\) involving multiplicative derivations. Moreover, someresults on one-sided ideals of \(R\) are given.

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