On matrix semiring over the extended tropical semiring

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