We investigate Sharifan and Moradi’s closed neighborhood ideal of a finite simple graph, which is square-free monomial in polynomial ring over field. explicitly describe the minimal irreducible decompositions these ideals. also characterize trees whose ideals are Cohen–Macaulay; particular, this property for characteristic independent.

In this paper we prove that if G is a group for which there are k non-Frattini chief factors isomorphic to characteristically simple A, then has normal section C/R the direct product of minimal subgroups G/R A. This significant extension notion crown factors.