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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.
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.
We describe the commutator invariant subgroups of a nonreduced abelian group. We find out when all commutator invariant subgroups of a separable group and an algebraically compact torsionfree group are fully invariant and describe the E-centers and E-commutants of these and some other groups.Let A be an abelian group. Denote the endomorphism ring of A by E(A), put A 1 = ∞ n=1 nA, and let r(A) be the rank of A. Unless otherwise specified, A p is the p-component, t(A) is the periodic part, andIf a is an element of order p k then by e(a) = k we denote the exponent of a. The expression H A means that H is a subgroup in A, and the expression H fiA, that H is a fully invariant subgroup in A, i.e., ϕH ⊆ H for every ϕ ∈ E(A). (B,G) fX be the subgroup in G generated by all homomorphic images of X. Denote by 1 A the identity endomorphism of a group A. If A is a homogeneous torsion-free group then t(A) is the type of A, N is the set of all natural numbers, Z is the ring or the group of integers, Q is the field of all rationals, Z p is the ring or the group of p-adic integers, and P is the set of all primes.AIf R is a ring then the operation ϕ • ψ = ϕψ − ψϕ (where ϕ, ψ ∈ R) is called commutation and the element [ϕ, ψ] = ϕψ − ψϕ is called the commutator of ϕ and ψ.The following are verified directly:(1) [α, β]γ = α[β, γ] + [αγ, β], [α, β]γ = [α, βγ] + β[γ, α]; (2) γ[α, β] = [γ, α]β + [α, γβ], γ[α, β] = [γα, β] + [β, γ]α; (3) in the ring R, the operation of commutation is associative if and only if every commutator of a ring R lies in the center of R. Indeed, [[a, b], c] = abc − bac − cab + cba, [a, [b, c]] = abc − acb − bca + cba. Equating the right-hand sides, we infer 0 = bac + cab − acb − bca = [[c, a], b], whence [c, a] ∈ Z(R). Call a subgroup H A commutator invariant (briefly, a ci-subgroup) if [ϕ, ψ]h ∈ H for all h ∈ H and ϕ, ψ ∈ E(A). Each subgroup in a group with commutative endomorphism ring is a ci-subgroup. Lemma 1. Let A = i∈I A i and let π i : A → A i be the corresponding projections and H A. Then (1) H ciA if and only if Hom(and only if Hom(A i , A j )B i ⊆ B j for all i, j ∈ I, where j = i; (3) if A i fiA and B i A i then B = i∈I B i ciA if and only if B i ciA i for all i ∈ I; (4) a ci-subgroup H of A is its fi-subgroup if and only if π i H = H ∩ A i and H ∩ A i fiA i for every i ∈ I.
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