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Let R be a ring with identity, X(R) the set of all nonzero, non-units of R and G(R) the group of all units of R. We show that for a matrix ring Mn(D), n ≥ 2, if a,b are singular matrices of the same rank, then |oℓ(a)| = |oℓ(b)|, where oℓ(a) and oℓ(b) are the orbits of a and b, respectively, under the left regular action. We also show that for a semisimple Artinian ring R such that X(R) ≠ ∅, R [수식], with Di infinite division rings of the same cardinalities or R is isomorphic to the ring of 2×2 matrices over a finite field if and only if |oℓ(x)| = |oℓ(y)| for all x,y ∈ X(R).
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참고문헌 (5)
참고문헌 신청
J. Cohen / 1989 / Half-transitive group actions in a compact ring / J. Pure Appl. Algebra 60 (2)
J. Han / 1997 / Regular action in a ring with a finite number of orbits / Comm. Algebra 25 (7) : 2227 ~ 2236
T. Y. Lam / 1991 / A First Course in Noncommutative Rings / Springer-Verlag
W. K. Nicholson / 1998 / Introduction to Abstract Algebra / PWS
Y. Hirano / 2004 / Rings with finitely many orbits under the regular action, Rings, modules, algebras, and abelian groups / Dekker : 343 ~ 347
J. Han / 1997 / Regular action in a ring with a finite number of orbits / Comm. Algebra 25 (7) : 2227 ~ 2236
T. Y. Lam / 1991 / A First Course in Noncommutative Rings / Springer-Verlag
W. K. Nicholson / 1998 / Introduction to Abstract Algebra / PWS
Y. Hirano / 2004 / Rings with finitely many orbits under the regular action, Rings, modules, algebras, and abelian groups / Dekker : 343 ~ 347
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