VDM McGuire, T: The Gröbner Annihilator Graph of a Ring

In recent years, graph theory has been used as a tool to study rings in the form of several different graphs, many of which are based on the zero divisor structure of the ring. We de ne here the annihilator graph of a ring to try to harness the best possible graphical representation of a ring. This paper lays the foundation for the theory of annihilator graphs and the extension of them to a more general form, the Gröbner annihilator graph. We will study the relationships between the algebraic properties of a ring, and the graph theoretic properties of the Gröbner annihilator graph of that ring. We will state some concise results while also formally stating areas in which the author feels further research is needed. This work is an extension of recent work on Zero Divisor Graphs, and Annihilator graphs, with an emphasis on simultaneous generalizations of both.