On distributive semimodules

Abstract

This work considers the construction of the concept of distributive property for semimodules. Some characterizations of this property, with some examples are given. Some conditions on semiring or semimodules (like subtractive, semisubtractive, cancellative, and k-cyclic) are required to obtain interesting results. The main results are: Any subsemimodule and factor semimodule of a distributive semimodule is distributive. Moreover, weakly distributive, is also, introduced and investigated. It is found that the semimodule is distributive if and only if each subsemimodule is weak distributive. Taking advantage of the supplemented concept to find some properties of distributive semimodule. Finally, the summand sum and summand intersection properties for distributive semimodules with some conditions are valid.

Distributive Semimodules: Theory, Properties, and Extensions

“…The following Proposition mentioned in [3],will be proved under different conditions. Proposition 3.7.…”

“…Let R be a commutative semiring with identity. An Rsemimodule M is said to be distributive if, for all subsemimodules 𝐴, 𝐵, and 𝐶 of 𝑀, the following equality holds: 𝐴 ∩ (𝐵 + 𝐶) = (𝐴 ∩ 𝐵) + ( 𝐴 ∩ 𝐶) [2].The notion of distributive semimodules has been studied and developed as a generalization independently in [2] and [3]. As for the module, in the last six decades much research and results on the structure of the modules with a distributive lattice of submodules (see for example [4], [5], [6], [7], and [8]).…”