Affiliation:
1. Department of Mathematics, Jadavpur University, Jadavpur, Kolkata 700032, India
2. Department of Mathematics, Techno India University, Kolkata 700091, India
Abstract
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).
Publisher
World Scientific Pub Co Pte Ltd