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The zero-divisor graph of a commutative ring
Author
Liaw, Valerie Bee Ling
Supervisor
Teo, Kok Ming
Abstract
For a commutative ring R, we can associate a zero-divisor graph T(R) to it. Then we investigate the relation between the ring theoretic properties of R with the graph theoretic properties of T(R).
Firstly, we focus on some examples of T(R) and their associated rings. We realize that some non-isomorphic rings produce the same T(R). From these examples, we also found that T(R) is of a certain form when the ring R satisfied certain conditions. For example, if a commutative ring R is finite or is an integral domain, then T(R) will also be finite. We also found some examples of graphs that cannot be realized as T(R).
Then we will explore some properties of T(R) that will help us to determine when certain classes of graphs can be realized. as T(R). We will look into the cases of when bipartite, star and complete graphs may be realized as T(R).
Firstly, we focus on some examples of T(R) and their associated rings. We realize that some non-isomorphic rings produce the same T(R). From these examples, we also found that T(R) is of a certain form when the ring R satisfied certain conditions. For example, if a commutative ring R is finite or is an integral domain, then T(R) will also be finite. We also found some examples of graphs that cannot be realized as T(R).
Then we will explore some properties of T(R) that will help us to determine when certain classes of graphs can be realized. as T(R). We will look into the cases of when bipartite, star and complete graphs may be realized as T(R).
Date Issued
2005
Call Number
QA251.3 Lia
Date Submitted
2005