Graphs and Zero-divisors

In an algebra class, one uses the zero-factor property to solve polynomial equations. For example, consider the equation x 2 = x. Rewriting it as x (x − 1) = 0, we conclude that the solutions are x = 0, 1. However, the same equation in a different number system need not yield the same solutions. For example, in Z 6 (the integers modulo 6), not only 0 and 1, but also 3 and 4 are solutions. (Check this!) In a commutative ring R (like Z 6), an element r is a zero-divisor if there exists a nonzero s ∈ R such that rs = 0. In Z 6 the zero-divisors are 0, 2, 3, and 4 because 0 · 2 = 2 · 3 = 3 · 4 = 0. A commutative ring with no nonzero zero-divisors is called an integral domain. The zero-factor property used in high school algebra holds for integral domains, but does not hold for all commutative rings. Because of this, ring theorists find zero-divisors very interesting. In general, the set of zero-divisors lacks algebraic structure. In particular, the set of all zero-divisors of a ring R, denoted Z(R), is not always closed under addition. In Z 6 , we see that 2 and 3 are zero-divisors, but 2+3 is not. Hence, Z(R) is typically not a subring and thus also not an ideal. Recently, a new approach to studying the set of zero-divisors has emerged from an unlikely direction: graph theory. In this paper we present a series of projects that develop the connection between commutative ring theory and graph theory. These are suitable for a student who has completed an introductory undergraduate abstract algebra course. Projects not marked with an asterisk are straightforward and should require well less than