On \phi-1-absorbing primary submodules
Ünsal Tekir\,^1 , Eda Yıldız\,^2, Suat Koç\,^3 and Ece Yetkin Çelikel\,^4
\,^1Department of Mathematics, Marmara University, Istanbul, Turkey.
\,^2Department of Mathematics, Yildiz Technical University, Istanbul, Turkey.
\,^3Deparment of Mathematics, Istanbul Medeniyet University, Istanbul, Turkey.
\,^4Department of Basic Sciences, Faculty of Engineering, Hasan Kalyoncu University, Gaziantep, Turkey.
Pages 232–242 | Received 20 November 2023, Accepted 27 January 2024, Published 08 November 2024
Abstract
In this article, we introduce \phi-1-absorbing primary submodules of modules over commutative rings. Let R\ be a commutative ring with a nonzero identity and M\ be a nonzero unital module. \phi:\mathcal{S}(M)\rightarrow \mathcal{S}(M)\cup{\emptyset} be a function where \mathcal{S}(M) is the set of all submodules of M. A proper submodule N\ of M\ is said to be a \phi-1-absorbing primary submodule if whenever abm\in N-\phi(N) for some nonunit elements a,b\in R\ and m\in M,\ then ab\in(N:_{R}M)\ or m\in M–rad(N),\ where M–rad(N) is the prime radical of N.\ Many properties and characterizations of \phi-1-absorbing primary submodules are given. We also give the relations between \phi-1-absorbing primary submodules and other classical submodules such as \phi-prime, \phi-primary, \phi -2-absorbing primary submodules.
Keywords: \phi-prime submodule, 1-absorbing primary submodule, (weakly) pseudo primary submodule, \phi-1-absorbing primary submodule.
MSC numbers: Primary 13A15; Secondary 16D60.
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