Uniqueness of primary decompositions in Laskerian le-modules

Abstract

Here we introduce and characterize a new class of le-modules R M where R is a commutative ring with 1 and (M, +, , e) is a lattice ordered semigroup with the greatest element e. Several notions are defined and uniqueness theorems for primary decompositions of a submodule element in a Laskerian le-module are established.

On the prime spectrum of an le-module

“…Then Ie is a submodule element of M. Also for any two ideals I and J of R, I ⊆ J implies that Ie Je. The following result, proved in [7], is useful here.…”

“…First we recall the definition of an le-module and various associated concepts from [7]. Here by an le-semigroup we mean (M, +, , e) such that (M, ≤) is a complete lattice, (M, +) is a commutative monoid with the zero element 0 M and for all m, m i ∈ M, i ∈ I it satisfies m ≤ e and…”

“…Since we are taking left action of a ring R not of the complete modular lattice of all ideals of R, we hope that influence of arithmetic of R on M will be easier to understand. For further details and motivation for introducing le-modules we refer to [7].…”

“…In addition to this introduction, this article comprises six sections. In Section 2, we recall definition of le-modules and various associated concepts from [7]. Also we discuss briefly on the Zariski topology in rings and modules.…”

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Here we continue to characterize a recently introduced notion, le-modules R M over a commutative ring R with unity [7]. This article introduces and characterizes Zariski topology on the set Spec(M ) of all prime submodule elements of M .

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