Distributive Semimodules: Theory, Properties, and Extensions

This article explores the properties and theorems pertaining to subtractive and distributive semimodules. The focus is on understanding the relations between these characteristics and their implications within the context of semimodules. The investigated propositions establish conditions for distributivity in semimodules. It is shown that under certain circumstances, the preimages of subsemimodules under homomorphisms exhibit specific additive properties. Furthermore, the study introduced definitions and propositions related to supplements, principally supplemented semimodules and H-supplemented semimodules. The findings reveal the importance of supplements in understanding the structure and properties of semimodules, particularly in relation to their distributive nature