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.

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.

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 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.

scholarly journals On periodic rings

2001 ◽  
Vol 25 (6) ◽  
pp. 417-420
Author(s):  
Xiankun Du ◽  
Qi Yi
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sum ◽  
Periodic Rings ◽  
Necessary And Sufficient ◽  
Nil Ring ◽  
Positive Integers

It is proved that a ring is periodic if and only if, for any elementsxandy, there exist positive integersk,l,m, andnwith eitherk≠morl≠n, depending onxandy, for whichxkyl=xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and aJ-ring.

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.

scholarly journals On Left Bipotent Near-Rings

1979 ◽  
Vol 22 (2) ◽  
pp. 99-107 ◽  
Author(s):  
J. L. Jat ◽  
S. C. Choudhary
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Singular Set ◽  
Direct Sum ◽  
Zero Divisors ◽  
Structure Theorems ◽  
Necessary And Sufficient ◽  
Regular Section

A near-ring N is defined to be left bipotent if Na = Na2 for each a in N. Many properties of such near-rings are proved in Section 1, and results of Chandran (4) are generalised. Most of the results are different from, and contrary to, the ring case. Necessary and sufficient conditions have also been obtained under which such near-rings become regular. Section 2 deals with left bipotent near-rings without zero divisors. Some structure theorems for direct sum decompositions and J(N) = (0) are proved and it is shown that for a left bipotent S-near-ring, the singular ‘set’ S(N) = 0. Necessary examples and counter examples are supplied.

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].

scholarly journals The lambda-property for generalised direct sums of normed spaces

1990 ◽  
Vol 41 (3) ◽  
pp. 441-450
Author(s):  
Robert H. Lohman ◽  
Thaddeus J. Shura
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normed Spaces ◽  
Reflexive Banach Spaces ◽  
Direct Sums ◽  
Necessary And Sufficient

This paper considers direct sums of normed spaces with respect to a Banach space with a normalised, unconditionally strictly monotone basis. Necessary and sufficient conditions are given for such direct sums to have the λ-property. These results are used to construct examples of reflexive Banach spaces U and V such that U has the uniform λ-property but U* fails to have the λ-property, while V and V* fail to have the λ-property.

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.

Constacyclic codes over the ring Fq[u,v,w]/〈u2 − 1, v2 − 1,w3 − w, uv − vu,vw − wv,wu − uw〉

2021 ◽  
Author(s):  
Shunhua Zhang
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Constacyclic Code ◽  
Direct Sum ◽  
Necessary And Sufficient

Let [Formula: see text] be the ring [Formula: see text], where [Formula: see text] for any odd prime [Formula: see text] and positive integer [Formula: see text]. In this paper, we study constacyclic codes over the ring [Formula: see text]. We define a Gray map by a matrix and decompose a constacyclic code over the ring [Formula: see text] as the direct sum of constacyclic codes over [Formula: see text], we also characterize self-dual constacyclic codes over the ring [Formula: see text] and give necessary and sufficient conditions for constacyclic codes to be dual-containing. As an application, we give a method to construct quantum codes from dual-containing constacyclic codes over the ring [Formula: see text].

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