PURE CLOSED SUBOBJECTS AND PURE QUOTIENT GOLDIE DIMENSION

2019 ◽  
Vol 41 (1) ◽  
pp. 49-57
Author(s):  
Mustafa Kemal Berktaş ◽  
Semra Doğruöz ◽  
Azime Tarhan
Keyword(s):  
Goldie Dimension

LINEARLY INDEPENDENT ELEMENTS IN N-GROUPS WITH FINITE GOLDIE DIMENSION

2005 ◽  
Vol 42 (3) ◽  
pp. 433-441 ◽  
Author(s):  
SATYANARAYANA BHAVANARI ◽  
SYAM PRASAD KUNCHAM
Keyword(s):  
Goldie Dimension ◽  
Linearly Independent

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 RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

2012 ◽  
Vol 54 (3) ◽  
pp. 605-617 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
NOYAN ER ◽  
NİL ORHAN ERTAŞ
Keyword(s):  
Prime Ideals ◽  
Cyclic Module ◽  
Direct Sums ◽  
Goldie Dimension ◽  
Direct Sum ◽  
Injective Modules ◽  
Dedekind Domains ◽  
Singular Ring

AbstractDedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

On dual goldie dimension

1993 ◽  
Vol 21 (2) ◽  
pp. 665-674 ◽  
Author(s):  
Seog Hoon Rim ◽  
Kazunari Takemori
Keyword(s):  
Goldie Dimension

Dual goldie dimension of certain extension rings

1982 ◽  
Vol 10 (20) ◽  
pp. 2223-2231 ◽  
Author(s):  
K. Varadarajan
Keyword(s):  
Goldie Dimension

scholarly journals A note on dual Goldie dimension

1991 ◽  
Vol 76 (2) ◽  
pp. 225-227
Author(s):  
Weimin Xue
Keyword(s):  
Goldie Dimension

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.

ON PRIME AND SEMIPRIME GOLDIE MODULES

2011 ◽  
Vol 04 (02) ◽  
pp. 321-334 ◽  
Author(s):  
Nguyen Van Sanh ◽  
S. Asawasamrit ◽  
K. F. U. Ahmed ◽  
Le Phuong Thao
Keyword(s):  
Goldie Dimension

A right R-module M is called a Goldie module if it has finite Goldie dimension and satisfies the ACC for M-annihilator submodules of M. In this paper, we study the class of prime Goldie modules and the class of semiprime Goldie modules as generalizations of prime right Goldie rings and semiprime right Goldie rings.

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