scholarly journals Direct Sums of Prime Subsemimodules

2022 ◽  
Vol 18 (2) ◽  
pp. 169-173
Author(s):  
Andi Muhammad Anwar ◽  
Hanni Garminia ◽  
Irawati Irawati
Keyword(s):  
Direct Sums ◽  
Commutative Semiring ◽  
Direct Sum ◽  
The Relationship

Let  be a commutative semiring. A semimodule  over a semiring  is  a fully prime semimodule if each proper subsemimodule of  is prime. This research aims to investigate the relationship between a direct sum of prime subsemimodules   and  ,  , and a fully prime semimodule.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Reduction to Dimension Two of the Local Spectrum for an AH Algebra with the Ideal Property

2017 ◽  
Vol 60 (4) ◽  
pp. 791-806 ◽  
Author(s):  
Chunlan Jiang
Keyword(s):  
Inductive Limit ◽  
Local Spectrum ◽  
Direct Sums ◽  
Matrix Algebras ◽  
Direct Sum ◽  
Ideal Property ◽  
Uniformly Bounded ◽  
The Ideal

AbstractA C*-algebra Ahas the ideal property if any ideal I of Ais generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limit C*-algebra A of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove that A can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsion H2 groups.

scholarly journals Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Brendan Goldsmith ◽  
Ketao Gong
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sums ◽  
Direct Sum ◽  
Zero Entropy ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

scholarly journals Relative injectivity and CS-modules

1994 ◽  
Vol 17 (4) ◽  
pp. 661-666
Author(s):  
Mahmoud Ahmed Kamal
Keyword(s):  
Direct Decomposition ◽  
Injective Hull ◽  
Extra Condition ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules ◽  
Semisimple Module ◽  
Continuous Module

In this paper we show that a direct decomposition of modulesM⊕N, withNhomologically independent to the injective hull ofM, is a CS-module if and only ifNis injective relative toMand both ofMandNare CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-injective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every finite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective.

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Moore Cohomology and Central Twisted Crossed Product C*-Algebras

1996 ◽  
Vol 48 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Judith A. Packer
Keyword(s):  
Cohomology Group ◽  
Hausdorff Space ◽  
Countable Group ◽  
Equivalence Classes ◽  
Locally Compact ◽  
C Algebras ◽  
Integer Coefficients ◽  
Direct Sum ◽  
Cech Cohomology ◽  
The Relationship

AbstractLet G be a locally compact second countable group, let X be a locally compact second countable Hausdorff space, and view C(X, T) as a trivial G-module. For G countable discrete abelian, we construct an isomorphism between the Moore cohomology group Hn(G, C(X, T)) and the direct sum Ext(Hn-1(G), Ȟl(βX, Ζ)) ⊕ C(X, Hn(G, T)); here Ȟ1 (βX, Ζ) denotes the first Čech cohomology group of the Stone-Čech compactification of X, βX, with integer coefficients. For more general locally compact second countable groups G, we discuss the relationship between the Moore group H2(G, C(X, T)), the set of exterior equivalence classes of element-wise inner actions of G on the stable continuous trace C*-algebra C0(X) ⊗ 𝒦, and the equivariant Brauer group BrG(X) of Crocker, Kumjian, Raeburn, and Williams. For countable discrete abelian G acting trivially on X, we construct an isomorphism is the group of equivalence classes of principal Ĝ bundles over X first considered by Raeburn and Williams.

Sur les Sous-Groupes Purifiables des Groupes Abéliens Primaires

1990 ◽  
Vol 33 (1) ◽  
pp. 11-17 ◽  
Author(s):  
K. Benabdallah ◽  
C. Piché
Keyword(s):  
Abelian Groups ◽  
Divisible Group ◽  
Direct Sums ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Direct Sum

AbstractThe class of primary abelian groups whose subsocles are purifiable is not yet completely characterized and it contains the class of direct sums of cyclic groups and torsion complete groups. In sharp constrast with this, the class of groups whose p2-bounded subgroups are purifiable consist only of those groups which are the direct sum of a bounded and a divisible group. Various tools are developed and a short application to the pure envelopes of cyclic subgroups is given in the last section.

scholarly journals Rings decomposed into direct sums ofJ-rings and nil rings

1985 ◽  
Vol 8 (1) ◽  
pp. 205-207 ◽  
Author(s):  
Hisao Tominaga
Keyword(s):  
Direct Consequence ◽  
Direct Sums ◽  
Direct Sum ◽  
Nil Ring

LetRbe a ring (not necessarily with identity) and letEdenote the set of idempotents ofR. We prove thatRis a direct sum of aJ-ring (every element is a power of itself) and a nil ring if and only ifRis stronglyπ-regular andEis contained in someJ-ideal ofR. As a direct consequence of this result, the main theorem of [1] follows.

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.

The Intersection Property of Annihilators of Both Modules and Maximal Submodules Belonging to the Same Module

2020 ◽  
Vol 17 (2) ◽  
pp. 552-555
Author(s):  
Hatam Yahya Khalaf ◽  
Buthyna Nijad Shihab
Keyword(s):  
Commutative Ring ◽  
Intersection Property ◽  
Direct Sum ◽  
Unitary Module ◽  
Maximal Submodule ◽  
The Relationship

During that article T stands for a commutative ring with identity and that S stands for a unitary module over T. The intersection property of annihilatoers of a module X on a ring T and a maximal submodule W of M has been reviewed under this article where he provide several examples that explain that the property. Add to this a number of equivalent statements about the intersection property have been demonstrated as well as the direct sum of module that realize that the characteristic has studied here we proved that the modules that achieve the intersection property are closed under the direct sum with a specific condition. In addition to all this, the relationship between the modules that achieve the above characteristics with other types of modules has been given.

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