Absorbing Elements in Lattice Modules

Abstract

Abstract. In this paper we introduce and investigate 2-absorbing, n-absorbing, (n, k)-absorbing, weakly 2-absorbing, weakly n-absorbing and weakly (n, k)-absorbing elements in a lattice module M . Some characterizations of 2-absorbing and weakly 2-absorbing elements of M are obtained. By counter example it is shown that a weakly 2-absorbing element of M need not be 2-absorbing. Finally we show that if N ∈ M is a 2-absorbing element, then rad(N ) is a 2-absorbing element of M .Mathematics Subject Classification (2010): 06D10, 06E10, 06E99, 06F99

Pseudo-Primary, Classical Prime and Pseudo-Classical Primary Elements in Lattice Modules

“…According to the definition 3.1 in [12], the radical of a proper element N ∈ M is denoted as rad(N ) and is defined as the element ∧{P ∈ M | P is a prime element and N P }. Clearly, if N ∈ M is itself prime then rad(N ) is also prime.…”

“…Obviously, N rad(N ) for all N ∈ M . From Theorem 3.6 of [12], it is clear that rad(N ) = N if N is either a radical element or a prime element of a multiplication lattice L-module M . Further, in any L-module M , by Lemma 3.5 of [12], it follows that √ N : I M (rad(N ) : I M ) and equality holds if M is a multiplication lattice L-module, as shown in the following theorem.…”

“…From Theorem 3.6 of [12], it is clear that rad(N ) = N if N is either a radical element or a prime element of a multiplication lattice L-module M . Further, in any L-module M , by Lemma 3.5 of [12], it follows that √ N : I M (rad(N ) : I M ) and equality holds if M is a multiplication lattice L-module, as shown in the following theorem. Obviously, (rad(N ) : I M ) = (N : I M ) if N is either a radical element of M or a prime element of a multiplication lattice L-module M .…”

φ-Prime and φ-Primary Elements in Lattice Modules

“…An L-module M is said to be Noetherian, if M satisfies the ascending chain condition, is modular and is principally generated. According to [17], a proper element Q of an L-module M is said to be 2-absorbing if for all a, b ∈ L, N ∈ M , abN Q implies either ab (Q : I M ) or bN Q or aN Q. According to [6], a proper element Q of an L-module M is said to be 2-absorbing primary if for all a, b…”

Uniqueness of primary decompositions in Laskerian le-modules

“…What follows, there are numerous publications studying in abstract form the class of submodules of a module over a commutative ring. Researchers considered a complete lattice M together with an action of a multiplicative lattice L on M similar to the action of a ring on an additive abelian group, as if M is the class of all subsets of an R-module and L is the class of all subsets of the ring R. Such a system M is known as L-module or lattice module [1,12] - [15,20,21,24,32,33].…”