On matrix semiring over the extended tropical semiring

In this paper, we first discuss about the basic semiring properties; subsemirings, ideals, [Formula: see text]-ideals, [Formula: see text]-ideals, units, structure preserving properties of both extended tropical semiring [Formula: see text] and semiring of tropical matrices [Formula: see text]. Using the known fact that the subsemimodules of [Formula: see text]-semimodule [Formula: see text] and the ideals of [Formula: see text] are in a canonical correspondence, we study different types of subsemimodules of [Formula: see text]-semimodule [Formula: see text], left (right) ideals, left (right) [Formula: see text]-ideals, left (right) [Formula: see text]-ideals of [Formula: see text]. We prove that both [Formula: see text] and [Formula: see text] are local semirings, Noetherian (Artinian), [Formula: see text]-Noetherian ([Formula: see text]-Artinian), [Formula: see text]-Noetherian ([Formula: see text]-Artinian). Finally, we deduce that [Formula: see text] is neither left nor right Noetherian (Artinian) but left as well as right [Formula: see text]-Noetherian ([Formula: see text]-Artinian).