Abstract
We obtain some elementary residuation properties in lattice modules and obtain a relation between a weakly primary element in a lattice module M and weakly prime element of a multiplicative lattice L.
Highlights
A multiplicative lattice L is a complete lattice provided with commutative, associative, and join distributive multiplication in which the largest element I acts as a multiplicative identity
A proper element p of L is said to be prime if ab ≤ p implies a ≤ p or b ≤ p
A proper element p of L is said to be primary if ab ≤ p implies a ≤ p or bn ≤ p for some positive integer n
Summary
A multiplicative lattice L is a complete lattice provided with commutative, associative, and join distributive multiplication in which the largest element I acts as a multiplicative identity. A proper element p of L is said to be prime if ab ≤ p implies a ≤ p or b ≤ p. A proper element p of L is said to be primary if ab ≤ p implies a ≤ p or bn ≤ p for some positive integer n. An element a ∈ L is called compact if a ⩽ ∨αbα implies a ⩽ bα1 ∨ bα2 ∨ ⋅ ⋅ ⋅ ∨ bαn for some finite subset {α1, α2, . Let F(L∗) denote the set of all filters of L. An L-module M is called a multiplication Lmodule if for every element N ∈ M there exists an element a ∈ L such that N = aIM.
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