Unit-duo rings and related graphs of zero divisors
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (16)
- S. Akbari, "On the zero-divisor graph of a commutative ring", J. Algebra, 274(2), pp.847~855 (2004)
- D. F. Anderson, "The zero-divisor graph of a commutative ring", J. Algebra, 217(2), pp.434~447 (1999)
- I. Beck, "Coloring of a commutative ring", J. Algebra, 116, pp.208~226 (1998)
- J. Cohen, "Half-transitive group action in a compact ring", J. Pure Appl. Algebra, 60(2), pp.139~153 (1989)
- F. DeMeyer, "Zero divisor graphs of semigroups", J. Algebra, 283(1), pp.190~198 (2005)
- R. Diestel, "Graph Theory" (1997), Springer-Verlag
- J. Han, "Regular action in a ring with a finite number of orbits", Comm. Algebra, 25(7), pp.2227~2236 (1997)
- 한준철, "DUO RING PROPERTY RESTRICTED TO GROUPS OF UNITS", 대한수학회지, 52(3), pp.489~501 (2015), 대한수학회
- J. Han, "On the structure of abelian rings", Front. Math. China
- J. Han, "Regular action in rings", Comm. Algebra, 42(2), pp.872~879 (2014)
- 한준철, "RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION", 대한수학회지, 51(4), pp.655~663 (2014), 대한수학회
- S. P. Redmond, "The zero-divisor graph of non-commutative ring", Internat. J. Commu-tative Rings, 1, pp.203~211 (2002)
- S. P. Redmond, "Structure in the zero-divisor graph of non-commutative ring", Houston J. Math, 30(2), pp.345~355 (2004)
- S. Spiroff, "A zero divisor graph determined by equivalence classes of zero divisors", Comm. Algebra, 39(7), pp.2338~2348 (2011)
- T. Wu, "On directed zero-divisor graphs of finite rings", Discrete Math, 296(1), pp.73~86 (2005)
- W. Xue, "Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings", Comm. Algebra, 20(9), pp.2777~2788 (1992)