Planar of Special Idealization Rings

Abstract

Let R(+)N be the idealization of the ring R by the R-module N. In this paper, we investigate when Γ(R(+)N) is a Planar graph where R is an integral domain and we investigate when Γ(Zn(+)Zm) is a Planar graph.

Further properties of L-closed topological spaces

“…⇒) By Kuratowski's Theorem concerning compact topological spaces [16]. If a subset F of X is a closed and X is a P-space, if a point y / ∈π x (F×Y), y∈V where V is an open subset of Y, then ( X×V)∩F=φ because X×{y}⊆( X × Y)\F [1]. Hence, π x (F×Y) )∩V=φ , that is, a subset π x (F) of X is closed [15].…”