THE ZERO-DIVISOR GRAPH UNDER A GROUP ACTION IN A COMMUTATIVE RING
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering Γ(R), the zero-divisor graph of R, under the regular action on X by G as follows:(1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of Γ(R) which is adjacent to every other vertex in Γ(R) if and only if R is a local ring or R ~ Z2 × F where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, J2, . . . , Jn,R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.
Abstract
Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering Γ(R), the zero-divisor graph of R, under the regular action on X by G as follows:(1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of Γ(R) which is adjacent to every other vertex in Γ(R) if and only if R is a local ring or R ~ Z2 × F where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, J2, . . . , Jn,R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.
참고문헌 (11)
- 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. II, Ideal theoretic methods in commutative algebra", Lecture Notes in Pure and Appl. Math., 220, pp.61~72 (2001), Dekker
- D. F. Anderson, "The zero-divisor graph of a commutative ring", J. Algebra, 217(2), pp.434~447 (1999)
- I. Beck, "Coloring of commutative rings", J. Algebra, 116(1), pp.208~226 (1988)
- F. DeMeyer, "Zero divisor graphs of semigroups", J. Algebra, 283(1), pp.190~198 (2005)
- R. Diestel, "Graph Theory" (1997), pringer-Verlag
- J. Han, "Regular action in a ring with a finite number of orbits", Comm. Algebra, 25(7), pp.2227~2236 (1997)
- J. Han, "Group actions in a unit-regular ring", Comm. Algebra, 27(7), pp.3353~3361 (1999)
- S. P. Redmond, "The zero-divisor graph of non-commutative ring", Internat. J. Commutative Rings, 1(4), pp.203~211 (2002)
- S. P. Redmond, "Structure in the zero-divisor graph of a noncommutative ring", Houston J. Math., 30(2), pp.345~355 (2004)
- T. Wu, "On directed zero-divisor graphs of finite rings", Discrete Math., 296(1), pp.73~86 (2005)