T-Extending Semimodule over Semiring

Themain aim of this research is to present and study several basic characteristics of the idea of t-extending semimodules. The semimodule
$F$
is said to be a t-extending semimodule if each t-closed sub-semimodule of
$F$
is t-essential in a direct summand of
$F$
. Hence, the behavior of the t-extending semimodule is considered. In addition, the relationship between the t-essential (t-closed) and essential (closed) has been studied and investigated as well. Finally, in this work, there are a number of results related to the t-extending property, which is one of the generalizations of extending property, (every extending is t-extending, while the converse is not true).