Executive Summary : | Artinian rings, also known as Noetherian rings of Krull dimension zero, are a well-studied class in Commutative Algebra and related areas. They hold significant importance in algebra and geometry due to their ability to reduce higher-dimensional Noetherian rings questions modulo a system of parameters. I. Beck's 1988 paper "Colorings of a Commtative Ring" introduced the concept of zero-divisor graph associated to a ring, which is a simple graph whose vertex set is the set all elements of R and two vertices are adjacent if their product in R is zero. This notion was later modified by Anderson and Livingston to include only non-zero zero-divisors, simplifying the graph structure. The theory holds promise and offers a rich interplay between algebraic aspects of commutative rings and graph theoretic aspects of associated graphs. The proposed research work aims to study graphs associated to Artinian rings, specifically Gorenstein Artinian rings. By the classical Artin-Wedderburn theorem, every Artinian ring can be decomposed as finite direct product of Artinian local rings. The proposed study of zero-divisors of Artinian rings includes studying multilinear forms associated to Artinian local rings (R, m) with m^{n-1} principal and m^n=0, zero-divisor graphs associated to Artinian Gorensetin local rings, and determining the automorphism groups of these graphs. |