UNIT-DUO RINGS AND RELATED GRAPHS OF ZERO DIVISORS

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. (Chec...

Zero Divisors and Commutativity of Rings

On zero-divisor graphs of quotient rings and complemented zero-divisor graphs

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...

Divisors of Zero in Matric Rings

Characterizations of Krull Rings with Zero Divisors

We show that a ring is a Krull ring if and only if every nonzero regular prime ideal contains a t-invertible prime ideal if and only if every proper regular principal ideal is quasi-equal to a product of prime ideals.