Some particular substructures of semimodules and semirings are studied in the paper. The author considers the minimal primary decompositions by generalizing some well-known results in the case of pseudomodules. The concept of Noetherianness with respect to such substructures, called g- Noetherianness is also studied. The main result is the following one: A pseudomodule is Noetherian if and only if it is a component of maximal type of a g-Noetherian strong semiband of pseudomodules.