The zero-divisor graph under group actions in a noncommutative ring
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Let R be a ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. First, we investigate some connected conditions of the zero-divisor graph Γ(R) of a noncommutative ring R as follows: (1) if Γ(R) has no sources and no sinks, then Γ(R) is connected and diameter of Γ(R), denoted by diam(Γ(R)) (resp. girth of Γ(R), denoted by g(Γ(R))) is equal to or less than 3; (2) if X is a union of finite number of orbits under the left (resp. right) regular action on X by G, then Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3, in addition, if R is local, then there is a vertex of Γ(R) which is adjacent to every other vertices in Γ(R); (3) if R is unitregular, then Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3. Next, we investigate the graph automorphisms group of Γ(Mat2(Zp)) where Mat2(Zp) is the ring of 2 by 2 matrices over the galois field Zp (p is any prime).
Abstract
Let R be a ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. First, we investigate some connected conditions of the zero-divisor graph Γ(R) of a noncommutative ring R as follows: (1) if Γ(R) has no sources and no sinks, then Γ(R) is connected and diameter of Γ(R), denoted by diam(Γ(R)) (resp. girth of Γ(R), denoted by g(Γ(R))) is equal to or less than 3; (2) if X is a union of finite number of orbits under the left (resp. right) regular action on X by G, then Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3, in addition, if R is local, then there is a vertex of Γ(R) which is adjacent to every other vertices in Γ(R); (3) if R is unitregular, then Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3. Next, we investigate the graph automorphisms group of Γ(Mat2(Zp)) where Mat2(Zp) is the ring of 2 by 2 matrices over the galois field Zp (p is any prime).
참고문헌 (14)
- S. Akbari, "On the zero-divisor graph of a commutative ring", Journal of Algebra, 274(2), pp.847~855 (2004)
- D. F. Anderson, "The zero-divisor graph of a commutative ring. II, Ideal theoretic methods in commutative algebra (Columbia, MO, 1999)", Lecture Notes in Pure and Applied Mathematics, 220, pp.61~72 (2001)
- D. F. Anderson, "The zero-divisor graph of a commutative ring", Journal of Algebra, 217(2), pp.434~447 (1999)
- I. Beck, "Coloring of commutative rings", Journal of Algebra, 116(1), pp.208~226 (1988)
- A. W. Chatters, "Rings with chain conditions, Research Notes in Mathematics, 44. Pitman (Advanced Publishing Program)" (1980)
- F. DeMeyer, "Zero divisor graphs of semigroups", Journal of Algebra, 283(1), pp.190~198 (2005)
- R. Diestel, "Graph Theory, Graduate Texts in Mathematics, Vol. 173" (1997), Springer-Verlag
- J. Han, "Regular action in a ring with a finite number of orbits", Communications in Algebra, 25(7), pp.2227~2236 (1997)
- J. Han, "Half-transitive group actions in a left Artinian ring", Kyungpook Mathematical Journal, 37(2), pp.297~303 (1997)
- J. Han, "Group actions in a unit-regular ring", Communications in Algebra, 27(7), pp.3353~3361 (1999)
- N. Ganesan, "Properties of rings with a finite number of zero divisors. II", Mathematische Annalen, 161, pp.241~246 (1965)
- S. P. Redmond, "The zero-divisor graph of a non-commutative ring, Commutative rings", pp.39~47 (2002), Nova Sci. Publ.
- S. P. Redmond, "Structure in the zero-divisor graph of a noncommutative ring", Houston Journal of Mathematics, 30(2), pp.345~355 (2004)
- T. Wu, "On directed zero-divisor graphs of finite rings", Discrete Mathematics, 296(1), pp.73~86 (2005)