Bikramjit Roy

Let [ ] be a commutative ring with unity, [ ] be the set of all ideals of [ ] and [ ] be a subset of [ ]. Cayley sum graph of ideals of [ ], denoted by [ ], is a simple undirected graph with vertex set [ ] and two distinct vertices [ ] and [ ] are adjacent ([ ]), whenever [ ] or [ ], for some ideal [ ] in [ ]. In this paper, we determine the girth, chromatic number, clique number, perfect code, and perfectness of the Cayley sum graphs [ ] and [ ] (where [ ] denotes the set of all nonzero maximal ideals of [ ]). Finally, we classify the family of rings whose Cayley sum graphs are split, co-graph, threshold, chordal and claw-free.

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).