On matrix semiring over the extended tropical semiring

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

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