E-solvable modules

Abstract

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.