New results related to a conjecture of Moore |
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Authors: | A. Bahlekeh Sh. Salarian |
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Affiliation: | 1. Department of Mathematics, Gonbade-Kavous University, 4971799151, Gonbade-Kavous, Iran 3. School of Mathematics, Institute for Research in Fundamental Science(IPM), P.O. Box: 19395-5746, Tehran, Iran 2. Department of Mathematics, University of Isfahan, P.O. Box: 81746-73441, Isfahan, Iran
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Abstract: | Let Γ be a group, Γ′ be a subgroup of Γ of finite index, and R be a ring with identity. Assume that M is an RΓ-module whose restriction to RΓ′ is projective. Moore’s conjecture: Assume that, for all ${x in (Gamma-Gamma^{prime})}$ , either there is an integer n such that ${1 neq x^{n} in Gamma^{prime}}$ or x has finite order and is invertible in R. Then M is also projective over RΓ. In this paper, we consider an analogue of this conjecture for injective modules. It turns out that the validity of the conjecture for injective modules implies the validity of it on projective and flat modules. It is also shown that the conjecture for injective modules is true whenever Γ belongs to Kropholler’s hierarchy ${{bf LH}mathfrak{F}}$ . In addition, assume that M is an RΓ-module whose restriction to RΓ′ is Gorenstein projective (resp. injective), it is proved that M is Gorenstein projective (resp. injective) over RΓ whenever Γ′ is a subgroup of Γ of finite index. |
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