An Element Weakly Primary to Another Element

Abstract

We introduce the concept “An element weakly primary to another element” and using this concept we have generalized some result proved by Manjarekar and Chavan (2004). It is shown that if is a family of elements weakly primary to a in L, then is weakly primary to a.

An element prime to and primary to another element in lattice modules

“…We introduce the concepts of an element prime to and an element primary to another element in lattice modules.This is a generalization of the concept introduced by C.S.Manjarekar and N.S.Chavan [4] in multiplicative lattices.…”

An Element $ϕ$-$δ$-Primary to another Element in Multiplicative Lattices

“…in [3]. Further, the notion of an element primary to another element in a multiplicative lattice L is introduced by C. S. Manjarekar and Nitin S. Chavan in [4]. In an attempt to unify these notions of an element prime to another element and an element primary to another element in a multiplicative lattice L under one frame, an element δ-primary to another element in a multiplicative lattice L is introduced by Ashok V. Bingi in [1].…”

“…Further, the concept of an element weakly prime to another element and an element weakly primary to another element in a multiplicative lattice L is introduced by C. S. Manjarekar and U. N. Kandale in [5]. To generalise these concepts, the study of an element φ-prime to another element and an element φ-primary to another element in a multiplicative lattice L is done by Ashok V. Bingi in [2].…”

“…According to [3], an element b ∈ L is said to be prime to a proper element p ∈ L if xb p implies x p where x ∈ L. According to [4], an element b ∈ L is said to be primary to a proper element p ∈ L if xb p implies x √ p where x ∈ L * . According to [5], an element b ∈ L is said to be weakly prime to a proper element p ∈ L if 0 = xb p implies x p where x ∈ L and an element b ∈ L is said to be weakly primary to a proper element p ∈ L if 0 = xb p implies x √ p where…”