scholarly journals Chang-Mundici construction of an enveloping unital lattice-group of a BL-algebra

10.29007/8hz9 ◽  
2018 ◽  
Author(s):  
Celestin Lele ◽  
Jean Bernard Nganou
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Lattice Group ◽  
Mv Algebra

For any BL-algebra L, we construct an associated lattice ordered Abelian group that coincides with the Chang’s l-group of an MV-algebra when the BL-algebra is an MV-algebra. We prove that the Chang’s group of the MV-center of any BL-algebra L is a direct summand in the above group. We also find a direct description of the complement of the Chang’s group of the MV-center in terms of the filter of dense elements of L. Finally, we compute some examples of the introduced group.

On Purifiable Subsocles of a Primary Abelian Group

1971 ◽  
Vol 23 (1) ◽  
pp. 48-57 ◽  
Author(s):  
John Irwin ◽  
James Swanek
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Pure Subgroup ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Interesting Connection

In this paper we shall investigate an interesting connection between the structure of G/S and G, where S is a purifiable subsocle of G. The results are interesting in the light of a counterexample by Dieudonné [3, p. 142] who exhibits a primary abelian group G, where G/S is a direct sum of cyclic groups, but G is not a direct sum of cyclic groups. Surprisingly, the assumption of the purifiability of S allows G to inherit the structure of G/S. In particular, we show that if G/S is a direct sum of cyclic groups and S supports a pure subgroup H, then G is a direct sum of cyclic groups and if is a direct summand of G which is of course a direct sum of cyclic groups. It is also shown that if G/S is a direct sum of torsion-complete groups and S supports a pure subgroup H, then G is a direct sum of torsion-complete groups and H is a direct summand of G, and is also a direct sum of torsion-complete groups.

scholarly journals Rings whose additive endomorphisms are ring endomorphisms

1992 ◽  
Vol 45 (1) ◽  
pp. 91-103 ◽  
Author(s):  
Manfred Dugas ◽  
Jutta Hausen ◽  
Johnny A. Johnson
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Present Article ◽  
Recent Article ◽  
Additive Group ◽  
Ring Endomorphism

A ring R is said to be an AE-ring if every endomorphism of its additive group R+ is a ring endomorphism. Clearly, the zero ring on any abelian group is an AE-ring. In a recent article, Birkenmeier and Heatherly characterised the so-called standard AE-lings, that is, the non-trivial AE-rings whose maximal 2-subgroup is a direct summand. The present article demonstrates the existence of non-standard AE-rings. Four questions posed by Birkenmeier and Heatherly are answered in the negative.

An Abelian Group as a Direct Summand of the Multiplication Group

2014 ◽  
Vol 197 (5) ◽  
pp. 587-589
Author(s):  
A. A. Agafonov ◽  
A. M. Sebeldin
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Multiplication Group

On the Decomposition of Groups

1969 ◽  
Vol 21 ◽  
pp. 762-768 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Finite Groups ◽  
Commutative Case ◽  
Direct Sum ◽  
Infinite Subgroup

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.

scholarly journals The Chinese Remainder Theorem for Strongly Semisimple MV-Algebras and Lattice-Groups

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Vincenzo Marra
Keyword(s):  
Abelian Group ◽  
Chinese Remainder Theorem ◽  
Order Unit ◽  
Finitely Generated ◽  
Mv Algebra ◽  
Mv Algebras

AbstractAn MV-algebra (equivalently, a lattice-ordered Abelian group with a distinguished order unit) is strongly semisimple if all of its quotients modulo finitely generated congruences are semisimple. All MV-algebras satisfy a Chinese Remainder Theorem, as was first shown by Keimel four decades ago in the context of lattice-groups. In this note we prove that the Chinese Remainder Theorem admits a considerable strengthening for strongly semisimple structures.

Modules and abelian groups with a bounded domain of injectivity

2018 ◽  
Vol 17 (06) ◽  
pp. 1850108 ◽  
Author(s):  
Yilmaz Mehmet Demirci
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Direct Summand ◽  
Commutative Ring ◽  
Bounded Domain ◽  
Abelian Groups ◽  
Prime Integer

In this work, impecunious modules are introduced as modules whose injectivity domains are contained in the class of all pure-split modules. This notion gives a generalization of both poor modules and pure-injectively poor modules. Properties involving impecunious modules as well as examples that show the relations between impecunious modules, poor modules and pure-injectively poor modules are given. Rings over which every module is impecunious are right pure-semisimple. A commutative ring over which there is a projective semisimple impecunious module is proved to be semisimple artinian. Moreover, the characterization of impecunious abelian groups is given. It states that an abelian group [Formula: see text] is impecunious if and only if for every prime integer [Formula: see text], [Formula: see text] has a direct summand isomorphic to [Formula: see text] for some positive integer [Formula: see text]. Consequently, an example of an impecunious abelian group which is neither poor nor pure-injectively poor is given so that the generalization defined is proper.

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