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).