By Leonid Kurdachenko, Javier Otal, Igor Ya Subbotin

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“The concept of modules over workforce jewelry RG for limitless teams G over arbitrary jewelry R is a truly large and complicated box of study with numerous scattered effects. … on account that the various effects look for the 1st time in a publication it may be prompt warmly to any professional during this box, but in addition for graduate scholars who're offered the wonderful thing about the interaction of the theories of teams, jewelry and representations.” (G. Kowol, Monatshefte für Mathematik, Vol. 152 (4), December, 2007)

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**Sample text**

Then A ≤ ζ(G). Proof. Suppose the contrary, that is ζ(G) contains no A. Then Aζ(G)/ζ(G) is a non-identity normal subgroup of the hypercentral group G/ζ(G), and hence there exists some a ∈ (A ∩ ζ2 (G)) \ ζ(G). Since A is a p-subgroup, we may assume that ap ∈ ζ(G). Then the mapping φ : g → [g, a], g ∈ G is an endomorphism of G such that [G, a] = Im φ = 1 since a ∈ ζ(G). Actually, since ap ∈ ζ(G), [G, a]p = 1 , and so Im φ is an elementary abelian p-subgroup. Since [G, a] = Im φ ∼ = G/ Ker φ = G/CG (a), we deduce that G/CG (a) is an elementary abelian p-group as well.

3. Let R be a ring, and let A be an R-module. Then the following statements are equivalent. (1) A is a sum of simple R-submodules. (2) A is a direct sum of simple R-submodules. (3) For every R-submodule B, there is an R-submodule C such that A = B ⊕ C. Proof. 1. (2) ⇒ (3). Let B be a non-zero R-submodule of A. 2, B = λ∈Λ Mλ , where Mλ is a simple R-submodule for each λ ∈ Λ. This means that the family {Mλ | λ ∈ Λ} is independent. Then there is a maximal independent family M, such that {Mλ | λ ∈ Λ} ⊆ M.

Since H = K∈L K and aF H = K∈L aF K, we deduce that aF H is further a simple F H-submodule. 4. If A is a simple ZG-module, then the underlying additive group of A either is a divisible torsion-free abelian group or is a p-elementary abelian group, for some prime p. In the ﬁrst case we may think of A as a QG-module whereas in the second one we think of A as an Fp G-module. 7 the following result. 8. Let G be an abelian group and H a periodic subgroup of G. If A is a simple ZG-module, then A is a semisimple ZG-module.