On (m, n)-semiprime submodules

Pekin, Ayten; Koç, Suat; Uğurlu, Emel Aslankarayiğit

doi:10.3176/proc.2021.3.05

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2024

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“…N is called a semiprime (resp. an (m, n)-semiprime [19]) submodule if for r ∈ R and b ∈ M, r m b ∈ N implies rb ∈ N (resp. r n b ∈ N).…”

Section: Introductionmentioning

confidence: 99%

(m, n)-Closed Submodules of Modules over Commutative Rings

Yetkin Celikel,

Khashan

2024

Preprint

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Let R be a commutative ring and m, n be positive integers. We define a proper submodule N of an R-module M to be (m,n)-closed if for r∈R and b∈M, r^{m}b∈N implies rⁿ∈(N:_{R}M) or b∈N. This class of submodules lies properly between the classes of prime and primary submodules. Many characterizations, properties and supporting examples concerning this class of submodules are provided. The notion of (m,n)-modules is introduced and characterized. Furhermore, the (m,n)-closed avoidance theorem is proved. Finally, the (m,n)-closed submodules in amalgamated modules are studied.

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“…N is called a semiprime (resp. an (m, n)-semiprime [19]) submodule if for r ∈ R and b ∈ M, r m b ∈ N implies rb ∈ N (resp. r n b ∈ N).…”

Section: Introductionmentioning

confidence: 99%