In semimodules over semirings theory, essential subsemimodules and essential monomorphisms of semimodules are studied differently by some authors. This paper deals with the extension of essential subsemimodules to semiessential subsemimodules following the pattern on essential submodules in module theory. We have defined and characterized many new notions such as semiessential subsemimodules, semiessential monomorphisms, quasiessential monomorphisms, pseudoessential monomorphisms and M complement subsemimodules. We compare the class of semiessential semimodules and the class of essential semimodules.

ring, modules, semirings, semimodules, substractive subsemimodules, cancellative subsemimodules, essential subsemimodules, essential monomorphisms.