Total graphs and the compressed intersection annihilator graph / Mayssa Abdelhamid Mahmoud Soliman ; Supervised Mohamed A. Elsayed , Nefertiti Megahed

Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics

Let R be a commutative ring with a non-zero identity, H a multiplicative prime subset of R, M an R-module and U a multiplicative prime subset of M. In this thesis, rst we discuss the following two basic questions on di erent graphs: Question 1: what are the properties of the graph that a ring homomorphism preserves? and Question 2: what are the properties of the graph that a module homomorphism preserves?. We discuss question 1 on the total graph of T(w(R)) and on the generalized total graph GTH(R). Besides, we discuss question 2 on total graph Tw(M) of a module M and its generalization, namely, the total graph GTU(M) of a module M with respect to multiplicative prime subsets U. Next we de ne a new graph IA(R), called the compressed intersection annihilator graph, and investigate some of its properties. This graph is a generalization of the torsion graph wR(R) and has some advantages over the torsion graph and some other graphs. We study classes of rings for which the equivalence between the set of zero-divisors Z(R) of R being an ideal and the completeness of IA(R) holds. As well, we show that if IA(R) is nite, then there exists a subring S of R such that IA(S) {u223C}= IA(R). In addition, we show that the graph IA(R) with at least three vertices is connected, and its diameter is less than or equal to three. Finally, we determine the properties of the graph in the cases when R is Zn the ring of integers modulo n, the direct product of integral domains, the direct product of Artinian local rings

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