On projective invariant subgroups of Abelian groups

Chekhlov, A. R.

doi:10.1007/s10958-009-9743-1

Cited by 11 publications

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“…In [3] and [4] some other special properties of projection invariant subgroups were considered and, in addition, when they are fully invariant (see [19] too). The results established there can also be applied successfully to the proof of Theorem 2.1.…”

Section: Basic Resultsmentioning

confidence: 99%

“…4. A nonzero group G is called a strongly IFI-group if either it has only trivial fully invariant subgroups, or all its nonzero fully invariant subgroups are isomorphic otherwise.Definition 5.A nonzero group G is called a strongly IC-group if either it has only trivial characteristic subgroups, or all its nonzero characteristic subgroups are isomorphic otherwise.…”

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confidence: 99%

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On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

Chekhlov

Danchev

2015

Communications in Algebra

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We introduce two classes of abelian groups which have either only trivial fully invariant subgroups or all their nontrivial (respectively nonzero) fully invariant subgroups are isomorphic, called IFI-groups and strongly IFI-groups, such that every strongly IFIgroup is an IFI-group, respectively. Moreover, these classes coincide when the groups are torsion-free, but are different when the groups are torsion as well as, surprisingly, mixed groups cannot be IFI-groups. We also study their important properties as our results somewhat contrast with those from [13] and [14].

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Section: Basic Resultsmentioning

confidence: 99%

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confidence: 99%

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

Chekhlov

Danchev

2015

Communications in Algebra

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“…For sufficiently wide classes of p-groups, this topic was treated in [2,28,33,38,43] and in other papers. The works [7,19,21,34,37,39] and others are devoted to the investigation of this question in torsion-free and mixed groups. Although the notion of a cotorsion group and its generalizations are studied sufficiently well (see [3,20,23,41]), little is known about the lattice of fully invariant subgroups of a cotorsion group.…”

Section: Full Transitivity Of Algebraically Compact Groupsmentioning

confidence: 99%

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

Kemoklidze

2014

J Math Sci

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“…Due to the choice of the submodule P we have (Ā + P ) ∩ T = ∅. Now if a n ∈Ā + P , then a n+1 ∈ [Ā + P Commutatorically invariant subgroups of Abelian groups were also studied in [15,17].…”

Section: Proposition 24mentioning

confidence: 99%

“…It is clear that if A ≤ ci M , then αA ≤ ci M for any α ∈ Z E(M ) . Commutatorically invariant subgroups of Abelian groups were studied in [11], and projectively invariant subgroups of Abelian groups, i.e., subgroups invariant with respect to projections were studied in [10,13,16].It is well known that the commutator [x, y] is a bilinear alternating function of x, y. Some other properties of commutators are presented in [11,12,14].…”

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E-solvable modules

Chekhlov

2012

J Math Sci

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E-nilpotent and E-solvable modules have been defined. Some properties of such modules have been proved. For instance, all direct summands of an E-nilpotent module are fully invariant, and the E-commutant of an E-solvable module is contained in the intersection of all maximal commutatorically invariant submodules. Necessary and sufficient conditions under which a finite length module is E-solvable have been found.All the considered rings are supposed to be associative with 1; all the modules are unitary. The modules are considered over a fixed ring. By E(M ) we denote the endomorphism ring of the module M , by 1 M we denote its identity endomorphism, and Z(R) is the center of the ring R. The designationand N are modules, Hom(M, N ) is their homomorphism group, and ∅ = H ⊆ M , then Hom(M, N )H denotes a submodule in N generated by all the subsets fH, where f runs over the group Hom(M, N ). H denotes a module's submodule generated by its subset H = ∅; Z is the additive group (or the ring) of integers; N is the set of all natural numbers.The concepts of nilpotency and solvability are important for the theory of noncommutative groups and algebras (see, for instance, [1, 2, 19, 23]). As is well known, if multiplication in the endomorphism ring E(M ) of a module M is replaced by the commutation operation ϕ • ψ = ϕψ − ψϕ, we obtain a Lie endomorphism ring L E(M ) of the module M . In this paper, we consider the action of L E(M ) on the modules M and, by analogy, define E-nilpotent and E-solvable modules. The author has not met these concepts in the literature. E-Center and E-CommutantWe recall that if R is a ring and a, b ∈ R, then the element [a, b] = ab−ba is said to be the commutator of the elements a and b; if a 1 , . . . , a n ∈ R, then [a 1 , . . . , a n ] = [a 1 , . . . , a n−1 ], a n .A submodule A of the module M is said to be commutatorically invariant (denoted as A ≤ ci M ) if [ϕ, ψ]A ⊆ A for all ϕ, ψ ∈ E(M ). It is clear that if A ≤ ci M , then αA ≤ ci M for any α ∈ Z E(M ) . Commutatorically invariant subgroups of Abelian groups were studied in [11], and projectively invariant subgroups of Abelian groups, i.e., subgroups invariant with respect to projections were studied in [10,13,16].It is well known that the commutator [x, y] is a bilinear alternating function of x, y. Some other properties of commutators are presented in [11,12,14]. (2) If a is an invertible element of a ring, then a [b, c] (3) If a ring R has no nonzero nilpotent elements and satisfies the identity [x 1 , . . . , x n+1 ] = 0 for some n ∈ N, then it is commutative.Proof. Properties (1) and (2) can be verified directly. Let us suppose that u = [a 1 , . . . , a n ] = 0, where a 1 , . . . , a n ∈ R, and let a = [a 1 , . . . , a n−1 ]. Then au = a(aa n − a n a) = [a, aa n ] ∈ Z(R) and so a n [a, aa n ] = [a, aa n ]a n . It follows that a n a[a, a n ] = aa n [a, a n ] or [a, a n ] 2 = 0. Therefore, u = [a, a n ] = 0. Contradiction.

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On projective invariant subgroups of Abelian groups

Cited by 11 publications

References 0 publications

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

On Abelian Groups Having All Proper Fully Invariant Subgroups Isomorphic

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

E-solvable modules

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