On φ-δ-Primary Submodules

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Ersoy, B. A., Tekir, Ü., Kaya, E., Bolat, M., & Koç, S. (2022). On ϕ-δ-Primary Submodules. Iranian Journal of Science and Technology, Transactions A: Science, 46(2), 421-427.Özet
In this paper, we introduce and study a new class of submodules which unify the concepts of prime and primary submodules. Let M be a unital module over a commutative ring R,ϕ:L(M)→L(M)∪{∅} be a reduction function and δ: L(R) → L(R) be an expansion function, where L(M) is the lattice of all submodules of M and L(R) is the lattice of all ideals of R. A proper submodule N of M is said to be a ϕ-δ-primary submodule of M if whenever am∈ N- ϕ(N) for some a∈ R and m∈ M, then either a∈ δ((N: M)) or m∈ N. Many properties, characterizations and examples of ϕ-δ-primary submodules are given.