Essential ideal of a matrix nearring and ideal related properties of graphs

In this paper, we consider matrix maps over a zero-symmetric right nearring $N$ with 1. We define the notions of essential ideal, superfluous ideal, generalized essential ideal of a matrix nearring and prove results which exhibit the interplay between these ideals and the corresponding ideals of the base nearring $N$. We discuss the combinatorial properties such as connectivity, diameter, completeness of a graph (denoted by $\mathcal{L}_{g}(H)$) defined on generalized essential ideals of a finitely generated module $H$ over $N$. We prove a characterization for $\mathcal{L}_{g}(H)$ to be complete. We also prove $\mathcal{L}_{g}(H)$ has diameter at-most 2 and obtain related properties with suitable illustrations.