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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.
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참고문헌 (16)
참고문헌 신청
S. Akbari / 2004 / On the zero-divisor graph of a commutative ring / J. Algebra 274(2):847~855
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217(2):434~447
I. Beck / 1998 / Coloring of a commutative ring / J. Algebra 116:208~226
J. Cohen / 1989 / Half-transitive group action in a compact ring / J. Pure Appl. Algebra 60(2):139~153
F. DeMeyer / 2005 / Zero divisor graphs of semigroups / J. Algebra 283(1):190~198
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217(2):434~447
I. Beck / 1998 / Coloring of a commutative ring / J. Algebra 116:208~226
J. Cohen / 1989 / Half-transitive group action in a compact ring / J. Pure Appl. Algebra 60(2):139~153
F. DeMeyer / 2005 / Zero divisor graphs of semigroups / J. Algebra 283(1):190~198
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