Bull. Korean Math. Soc. 2016; 53(6): 1629-1643
Online first article August 25, 2016 Printed November 30, 2016
https://doi.org/10.4134/BKMS.b150684
Copyright © The Korean Mathematical Society.
Juncheol Han, Yang Lee, and Sangwon Park
Pusan National University, Pusan National University, Dong-A University
Let $R$ be a ring with identity, $X$ be the set of all nonzero, nonunits of $R$ and $G$ be the group of all units of $R$. A ring $R$ is called $unit$-$duo$ $ring$ if $[x]_{\ell} = [x]_{r}$ for all $x \in X$ where $[x]_{\ell} = \{ux \,|\, u \in G\}$ (resp. $[x]_{r} = \{xu \,|\, u \in G\}$) which are equivalence classes on $X$. It is shown that for a semisimple unit-duo ring $R$ (for example, a strongly regular ring), there exist a finite number of equivalence classes on $X$ if and only if $R$ is artinian. By considering the zero divisor graph (denoted $\widetilde{\Gamma} (R)$) determined by equivalence classes of zero divisors of a unit-duo ring $R$, it is shown that for a unit-duo ring $R$ such that $\widetilde{\Gamma} (R)$ is a finite graph, $R$ is local if and only if diam($\widetilde{\Gamma}(R)$) = 2.
Keywords: graph-designable ring, zero divisor graph
MSC numbers: Primary 05C20, 16W22
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