On ϕ-Absorbing Primary Elements in Lattice Modules

Sachin Ballal,Vilas Kharat

doi:10.1155/2015/183930

Abstract

Let L be a C-lattice and let M be a lattice module over L. Let ϕ:M→M be a function. A proper element P∈M is said to be ϕ-absorbing primary if, for x1,x2,…,xn∈L and N∈M, x1x2⋯xnN≤P and x1x2⋯xnN≰ϕ(P) together imply x1x2⋯xn≤(P:1M) or x1x2⋯xi-1xi+1⋯xnN≤PM, for some i∈{1,2,…,n}. We study some basic properties of ϕ-absorbing primary elements. Also, various generalizations of prime and primary elements in multiplicative lattices and lattice modules as ϕ-absorbing elements and ϕ-absorbing primary elements are unified.

Highlights

A lattice L is said to be complete, if, for any subset S of L, we have ∨S, ∧S ∈ L
An element p ≠ 1 of a multiplicative lattice L is said to be prime if ab ≤ p implies either a ≤ p or b ≤ p, for a, b ∈ L
An element p < 1 of a multiplicative lattice L is called n-absorbing if x1x2x3 ⋅ ⋅ ⋅ xn+1 ≤ p implies x1x2 ⋅ ⋅ ⋅ xi−1xi+1 ⋅ ⋅ ⋅ xn+1 ≤ p and called weakly n-absorbing if, for x1, x2, x3, . . . , xn+1 ∈ L, 0 ≠ x1x2x3 ⋅ ⋅ ⋅ xn+1 ≤ p implies x1x2 ⋅ ⋅ ⋅ xi−1xi+1 ⋅ ⋅ ⋅ xn+1 ≤ p, i ∈ {1, 2, . . . , n}, and x1, x2, x3, . . . , xn+1 ∈ L

Summary

Introduction

A lattice L is said to be complete, if, for any subset S of L, we have ∨S, ∧S ∈ L. A proper element P ∈ M is said to be φ-absorbing primary if, for x1, x2, .

Results

Conclusion