Zero Divisor Graphs of Amalgamations of Algebras along Ideals

In this project, we will study the zero divisor graphs of amalgamations of algebras along ideals. Let $R := A\bowtie^{f{ J$ be the amalgamation of commutative rings $A$ and $B$ along the ideal $J$ with respect to the ring homomorphism $f : A\longright B$. Our aim is to characterize when the zero divisor graph of $R$, denoted $\Gmma(R)$, is a complete graph, and when its diameter is equal to $2$ or $3$. We also investigate the girth $gr(\Gamma(R))$ of $\Gamma(R)$. All along this work, we will test the validity of some well-known results (with respect to the above properties) for rings with zero-divisors.