# Semimodules over commutative semirings and modules over unitary commutative rings

## TL;DR

This paper explores the structure of subsemimodules and submodules over commutative semirings and rings, establishing lattice and poset descriptions, and revealing a bijective correspondence with projections in modules.

## Contribution

It provides a detailed analysis of subsemimodule and submodule lattices, introducing new descriptions and a bijective link with projections in modules over commutative rings.

## Key findings

- Lattices of subsemimodules and submodules are characterized.
- Posets of splitting subsemimodules and submodules are described.
- A bijective correspondence between these posets and projections is established.

## Abstract

We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and submodules. In the case of modules a natural bijective correspondence between these posets and posets of projections is established.