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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.
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참고문헌 (11)
참고문헌 신청
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217 (2) : 434 ~ 447
D. F. Anderson / 2001 / The zero-divisor graph of a commutative ring. II, Ideal theoretic methods in commutative algebra / Lecture Notes in Pure and Appl. Math. 220 : 61 ~ 72
F. DeMeyer / 2005 / Zero divisor graphs of semigroups / J. Algebra 283 (1) : 190 ~ 198
I. Beck / 1988 / Coloring of commutative rings / J. Algebra 116 (1) : 208 ~ 226
J. Han / 1997 / Regular action in a ring with a finite number of orbits / Comm. Algebra 25 (7) : 2227 ~ 2236
D. F. Anderson / 2001 / The zero-divisor graph of a commutative ring. II, Ideal theoretic methods in commutative algebra / Lecture Notes in Pure and Appl. Math. 220 : 61 ~ 72
F. DeMeyer / 2005 / Zero divisor graphs of semigroups / J. Algebra 283 (1) : 190 ~ 198
I. Beck / 1988 / Coloring of commutative rings / J. Algebra 116 (1) : 208 ~ 226
J. Han / 1997 / Regular action in a ring with a finite number of orbits / Comm. Algebra 25 (7) : 2227 ~ 2236
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