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

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.

Download Full-text

Divisibility of Direct Sums in Torsion Theories

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-027-6 ◽

1976 ◽

Vol 28 (1) ◽

pp. 211-214 ◽

Cited By ~ 1

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.

Download Full-text

On multilattice groups. II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100039682 ◽

1966 ◽

Vol 62 (2) ◽

pp. 149-164 ◽

Cited By ~ 3

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.

Download Full-text

Partially Coherent Direct Sum Channels

Quantum ◽

10.22331/q-2021-07-15-504 ◽

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.

Download Full-text

On periodic rings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201001181 ◽

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.

Download Full-text

On Left Bipotent Near-Rings

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500016217 ◽

1979 ◽

Vol 22 (2) ◽

pp. 99-107 ◽

Cited By ~ 5

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.

Download Full-text

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

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500031 ◽

2019 ◽

Vol 19 (12) ◽

pp. 2150003 ◽

Cited By ~ 2

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].

Download Full-text

On Moufang loops that are abelian groups over the nucleus

International Journal of Algebra and Computation ◽

10.1142/s021819671550023x ◽

2015 ◽

Vol 25 (05) ◽

pp. 889-897

Author(s):

Piroska Csörgő ◽

Maria L. Merlini Giuliani

Keyword(s):

Abelian Group ◽

Automorphism Group ◽

Abelian Groups ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Moufang Loop ◽

Moufang Loops ◽

Necessary And Sufficient

We give some necessary and sufficient conditions for the equivalency of the following two properties for a Moufang loop Q: Q over the nucleus Q/N is an abelian group and Q over the center is a group. We study those properties of Moufang loops which guarantee that L(y, x) is in the automorphism group of the loop. These imply numerous statements, among them there are well-known old results.

Download Full-text

On polygonal products of finitely generated abelian groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700030343 ◽

1992 ◽

Vol 45 (3) ◽

pp. 453-462 ◽

Cited By ~ 10

Author(s):

Goansu Kim

Keyword(s):

Finite Groups ◽

Cyclic Subgroup ◽

Abelian Groups ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Finitely Generated ◽

Residually Finite ◽

Necessary And Sufficient

We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

Download Full-text

The lambda-property for generalised direct sums of normed spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700018323 ◽

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.

Download Full-text

On the structure of symmetric n-ary bands

International Journal of Algebra and Computation ◽

10.1142/s0218196721500478 ◽

2021 ◽

pp. 1-20

Author(s):

Jimmy Devillet ◽

Pierre Mathonet

Keyword(s):

Structure Theorem ◽

Abelian Groups ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Semilattice Decomposition ◽

Strong Semilattice ◽

Necessary And Sufficient

We study the class of symmetric [Formula: see text]-ary bands. These are [Formula: see text]-ary semigroups [Formula: see text] such that [Formula: see text] is invariant under the action of permutations and idempotent, i.e., satisfies [Formula: see text] for all [Formula: see text]. We first provide a structure theorem for these symmetric [Formula: see text]-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong [Formula: see text]-ary semilattice of [Formula: see text]-ary semigroups and we show that the symmetric [Formula: see text]-ary bands are exactly the strong [Formula: see text]-ary semilattices of [Formula: see text]-ary extensions of Abelian groups whose exponents divide [Formula: see text]. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric [Formula: see text]-ary band to be reducible to a semigroup.

Download Full-text