Lifting modules with finite internal exchange property and direct sums of hollow modules

2020 ◽  
pp. 2150189
Author(s):  
Yosuke Kuratomi
Keyword(s):  
Direct Summand ◽  
Sufficient Conditions ◽  
Exchange Property ◽  
Direct Sums ◽  
Perfect Ring ◽  
Direct Sum ◽  
Decomposition Formula ◽  
Direct Sum Decomposition ◽  
Extending Module ◽  
Necessary And Sufficient

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

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.

Quantum codes from a class of constacyclic codes over finite commutative rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2150003 ◽  
Author(s):  
Hai Q. Dinh ◽  
Tushar Bag ◽  
Ashish K. Upadhyay ◽  
Mohammad Ashraf ◽  
Ghulam Mohammad ◽  
...  
Keyword(s):  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Negacyclic Codes ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Necessary And Sufficient

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

Divisibility of Direct Sums in Torsion Theories

1976 ◽  
Vol 28 (1) ◽  
pp. 211-214 ◽  
Author(s):  
B. Sarath ◽  
K. Varadarajan
Keyword(s):  
Sufficient Conditions ◽  
Torsion Theory ◽  
Necessary And Sufficient Conditions ◽  
Countable Subset ◽  
Direct Sums ◽  
Direct Sum ◽  
Hereditary Torsion Theory ◽  
Torsion Theories ◽  
Necessary And Sufficient

Given a hereditary torsion theory on the category Mod R of right R-modules we obtain in this paper necessary and sufficient conditions for the direct sum of a given family of R-modules to be divisible for the torsion theory . Using this criterion we show that if is a family of R-modules having the property that is divisible for every countable subset K ol J then is itself divisible.

On multilattice groups. II

1966 ◽  
Vol 62 (2) ◽  
pp. 149-164 ◽  
Author(s):  
D. B. Mcalister
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Positive Element ◽  
Necessary And Sufficient Conditions ◽  
Countable Number ◽  
Direct Sums ◽  
Lattice Group ◽  
Ordered Groups ◽  
Group A ◽  
Necessary And Sufficient

Conrad ((2)), has shown that any lattice group which obeys (C.F.) each strictly positive element exceeds at most a finite number of pairwise orthogonal elements may be constructed, from a family of simply ordered groups, by carrying out, alternately, the operations of forming finite direct sums and lexico extensions, at most a countable number of times. The main result of this paper, Theorem 3.1, gives necessary and sufficient conditions for a multilattice group, which obeys (ℋ*), to be isomorphic to a multilattice group which is constructed from a family of almost ordered groups, by carrying out, alternately, the operations of forming arbitrary direct sums and lexico extensions, any number of times; we call such a group a lexico sum of the almost ordered groups.

scholarly journals Direct-sum decomposition of atomic and orthogonally complete rings

1970 ◽  
Vol 11 (3) ◽  
pp. 357-361 ◽  
Author(s):  
Alexander Abian
Keyword(s):  
Natural Number ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Image Position ◽  
Necessary And Sufficient

In this paper we give a necessary and sufficient condition for decomposition (as a direct sum of fields) of a ring R in which for every x ∈ R there exists a (and hence the smallest) natural number n(x) > 1 such that . We would like to emphasize that in what follows R stands for a ring every element x of which satisfies (1).

scholarly journals Partially Coherent Direct Sum Channels

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 504
Author(s):  
Stefano Chessa ◽  
Vittorio Giovannetti
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Structure ◽  
Quantum Channels ◽  
Amplitude Damping ◽  
Direct Sum ◽  
Partially Coherent ◽  
Quantum Capacity ◽  
Necessary And Sufficient

We introduce Partially Coherent Direct Sum (PCDS) quantum channels, as a generalization of the already known Direct Sum quantum channels. We derive necessary and sufficient conditions to identify the subset of those maps which are degradable, and provide a simplified expression for their quantum capacities. Interestingly, the special structure of PCDS allows us to extend the computation of the quantum capacity formula also for quantum channels which are explicitly not degradable (nor antidegradable). We show instances of applications of the results to dephasing channels, amplitude damping channels and combinations of the two.

Strongly Gorenstein categories

2021 ◽  
pp. 2250213
Author(s):  
Wan Wu ◽  
Zenghui Gao
Keyword(s):  
Direct Summand ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Abelian Category ◽  
Direct Products ◽  
Direct Sums ◽  
Gorenstein Categories

We introduce and study strongly Gorenstein subcategory [Formula: see text], relative to an additive full subcategory [Formula: see text] of an abelian category [Formula: see text]. When [Formula: see text] is self-orthogonal, we give some sufficient conditions under which the property of an object in [Formula: see text] can be inherited by its subobjects and quotient objects. Then, we introduce the notions of one-sided (strongly) Gorenstein subcategories. Under the assumption that [Formula: see text] is closed under countable direct sums (respectively, direct products), we prove that an object is in right (respectively, left) Gorenstein category [Formula: see text] (respectively, [Formula: see text]) if and only if it is a direct summand of an object in right (respectively, left) strongly Gorenstein subcategory [Formula: see text] (respectively, [Formula: see text]). As applications, some known results are obtained as corollaries.

Weyl’s theorem for direct sums

2007 ◽  
Vol 44 (2) ◽  
pp. 275-290
Author(s):  
Bhagwati Duggal ◽  
Carlos Kubrusly
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
The Other ◽  
Weyl’S Theorem ◽  
Direct Sums ◽  
Direct Sum ◽  
Hilbert Space Operators ◽  
Isolated Points

Let T and S be Hilbert space operators such that Weyl’s theorem holds for both of them. In general, it does not follow that Weyl’s theorem holds for the direct sum T ⊕ S . We give asymmetric sufficient conditions on T and S to ensure that the direct sum T ⊕ S satisfies Weyl’s theorem. It is assumed that just one of the direct summands satisfies Weyl’s theorem but is not necessarily isoloid, while the other has no isolated points in its spectrum.

scholarly journals On indecomposable decompositions of CS-modules

1996 ◽  
Vol 61 (1) ◽  
pp. 30-41 ◽  
Author(s):  
Nguyen V. Dung
Keyword(s):  
Direct Summand ◽  
Infinite Sequence ◽  
Exchange Property ◽  
Endomorphism Rings ◽  
Direct Sum

AbstractIt is shown that, over any ring R, the direct sum M = ⊕i∈IMi of uniform right R-modules Mi with local endomorphism rings is a CS-module if and only if every uniform submodule of M is essential in a direct summand of M and there does not exist an infinite sequence of non-isomorphic monomorphisms , with distinct in ∈ I. As a consequence, any CS-module which is a direct sum of submodules with local endomorphism rings has the exchange property.

A note on dual automorphism invariant modules

2017 ◽  
Vol 16 (02) ◽  
pp. 1750024
Author(s):  
C. Selvaraj ◽  
S. Santhakumar
Keyword(s):  
Projective Module ◽  
Exchange Property ◽  
Sufficient Condition ◽  
Cyclic Module ◽  
Necessary And Sufficient Condition ◽  
Perfect Ring ◽  
Necessary And Sufficient

In this paper, we investigate some properties of dual automorphism invariant modules over right perfect rings. Also, we introduce the notion of dual automorphism invariant cover and prove the existence of dual automorphism invariant cover. Moreover, we give the necessary and sufficient condition for every cyclic module to be a dual automorphism invariant module over a semi perfect ring and we prove that supplemented quasi projective module has finite exchange property. Also we give a characterization of a perfect ring using dual automorphism invariant module.

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