ϕ-Prime and ϕ-Primary Elements in Multiplicative Lattices

Abstract

We investigateϕ-prime andϕ-primary elements in a compactly generated multiplicative latticeL. By a counterexample, it is shown that aϕ-primary element inLneed not be primary. Some characterizations ofϕ-primary andϕ-prime elements inLare obtained. Finally, some results for almost prime and almost primary elements inLwith characterizations are obtained.

φ-Prime and φ-Primary Elements in Lattice Modules

“…In multiplicative lattices, the study of φ-prime and φ-primary elements is done by C. S. Manjarekar and A. V. Bingi in [16]. Our aim is to extend the notion of φ-prime and φ-primary elements in a multiplicative lattice to the notion of φ-prime and φ-primary elements in a lattice module and study its properties.…”

“…A proper element a ∈ L is said to be nilpotent if a n = 0 for some n ∈ Z + . According to [9], a proper element p ∈ L is said to be almost prime if for all a, b ∈ L, ab p and ab p 2 implies either a p or b p and according to [15], a proper element p ∈ L is said to be almost primary if for all a, b ∈ L, ab p and ab p 2 implies either a p or b √ p. Further study on almost prime and almost primary elements of a multiplicative lattice L is seen in [16], [5] and [4]. According to [12], a proper element q ∈ L is said to be 2-absorbing if for all a, b, c ∈ L, abc q implies either ab q or bc q or ca q.…”

“…According to [12], a proper element q ∈ L is said to be 2-absorbing if for all a, b, c ∈ L, abc q implies either ab q or bc q or ca q. According to [18], a proper element q ∈ L is said to be 2-absorbing primary if for all a, b, c ∈ L, abc q implies either ab q or bc √ q or ca √ q. The reader is referred to [2], [3] and [9] for general background and terminology in multiplicative lattices.…”

Some properties of 2-absorbing primary ideals in lattices

“…Manjarekar and Bingi [5] introduced 2-absorbing primary elements in multiplicative lattices. They defined, a proper element q ∈ L to be a 2-absorbing primary if for every a, b, c ∈ L , abc ≤ q implies either ab ≤ q or bc ≤ √ q or ca ≤ √ q. Celikel et al [6,7] introduced and studied φ-2-absorbing elements in multiplicative lattices.…”

On ϕ-Absorbing Primary Elements in Lattice Modules

“…Manjarekar and Bingi [6] unified the theory of generalizations of prime and primary elements in multiplicative lattice as -prime and -primary elements. Let : → be a function.…”