On submodule characterization and decomposition of modules over group rings

Uc, Mehmet; Alkan, Mustafa

doi:10.1063/1.4992475

Cited by 2 publications

(4 citation statements)

References 9 publications

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“…𝑀𝐺 and 𝑒̃𝐻. 𝑀𝐺 ≃ 𝑀(𝐺/𝐻) by the theorem in(Uc & Alkan, 2017). Since 𝑒̃𝐻 is a central idempotent by Lemma 14, we get 𝑀𝑆 = 𝑒̃𝐻.…”

mentioning

confidence: 81%

On Modules over G-sets

Uc,

Alkan

2023

jmss

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Let R be a commutative ring with unity, M a module over R and let S be a G–set for a finite group G. We define a set MS to be the set of elements expressed as the formal finite sum of the form ∑s∈Smss where ms∈M. The set MS is a module over the group ring RG under the addition and the scalar multiplication similar to the RG–module MG. With this notion, we not only generalize but also unify the theories of both of the group algebra and the group module, and we also establish some significant properties of (MS)RG. In particular, we describe a method for decomposing a given RG–module MS as a direct sum of RG–submodules. Furthermore, we prove the semisimplicity problem of (MS)RG with regard to the properties of MR, S and G.

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“…𝑀𝐺 and 𝑒̃𝐻. 𝑀𝐺 ≃ 𝑀(𝐺/𝐻) by the theorem in(Uc & Alkan, 2017). Since 𝑒̃𝐻 is a central idempotent by Lemma 14, we get 𝑀𝑆 = 𝑒̃𝐻.…”

mentioning

confidence: 81%

On Modules over G-sets

Uc,

Alkan

2023

jmss

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“…The theme of this section is the examination of a G-set module (M S) RG through the study of a decomposition of it. The decompositions of RG and (M G) RG obtained from the idempotent defined as e H = Ĥ |H| , where |H| is the order of H and Ĥ = h∈H h, explained in [11] and [15], respectively. A similar method give a criterion for the decomposition of a G-set module (M S) RG .…”

Section: The Decomposition Of (Ms) Rgmentioning

confidence: 99%

“…In examination of the studies in group rings which make use of the theory of group modules (see [4], [9], [15]), the semisimplicity problem of the G-set module arises. In [4], the generalized Maschke's Theorem states that a group ring RG a semisimple Artinian ring if and only if R is a semisimple Artinian ring, G is finite and |G| −1 ∈ R. A module theoretic version of the Maschke's Theorem is proven in [9].…”

Section: The Decomposition Of (Ms) Rgmentioning

confidence: 99%

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On Modules over a G-set

Uc,

Alkan

2017

Preprint

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Let R be a commutative ring with unity, M a module over R and let S be a G-set for a finite group G. We define a set M S to be the set of elements expressed as the formal finite sum of the form s∈S mss where ms ∈ M . The set M S is a module over the group ring RG under the addition and the scalar multiplication similar to the RG-module M G defined by Kosan, Lee and Zhou in [9]. With this notion, we not only generalize but also unify the theories of both of the group algebra and the group module, and we also establish some significant properties of (M S)RG. In particular, we describe a method for decomposing a given RG-module M S as a direct sum of RG-submodules. Furthermore, we prove the semisimplicity problem of (M S)RG with regard to the properties of MR, S and G.

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On submodule characterization and decomposition of modules over group rings

Cited by 2 publications

References 9 publications

On Modules over G-sets

On Modules over G-sets

On Modules over a G-set

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