On generalization of prime, weakly prime and almost prime elements in multiplicative lattices

Abstract

In this paper, we study the concepts of n-absorbing, weakly nabsorbing and n-almost n-absorbing elements as a generalization of prime, weakly prime and almost prime elements in C-lattices.

φ-Prime and φ-Primary Elements in Lattice Modules

“…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.…”

Absorbing Elements in Lattice Modules

“…al. in [6]. Our aim is to extend the notion of absorbing elements in a multiplicative lattice to a notion of absorbing elements in lattice modules and study its properties.…”

“…Obviously a prime element of L is a 1-absorbing element and a weakly prime element of L is a weakly 1-absorbing element. According to [6], a proper element of q ∈ L is said to be a n-absorbing element if for every…”

On ϕ-Absorbing Primary Elements in Lattice Modules

“…An element ̸ = 1 in is said to be weakly prime if 0 ̸ = ≤ implies either ≤ or ≤ and almost prime if ≤ and ≰ 2 implies either ≤ or ≤ for , ∈ . In [2], the authors generalized these concepts, respectively, to weakly primary and almost primary elements in multiplicative lattices. A proper element ∈ is said to be weakly primary if 0 ̸ = ≤ implies either ≤ or ≤√ and almost primary if ≤ and ≰ 2 implies either ≤ or ≤√ for , ∈ .…”

“…In [5], the authors defined the concepts of -absorbing, weakly -absorbing, and -almost -absorbing elements in multiplicative lattices as generalizations of, respectively, 2-absorbing, weakly 2-absorbing, and almost prime elements, where ≥ 2. An element < 1 of a multiplicative lattice is called -absorbing if 1 …”