RINGS WITH A FINITE NUMBER OF ORBITS UNDER THE REGULAR ACTION
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초록
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).
Abstract
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).
참고문헌 (5)
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- J. Han, "Regular action in a ring with a finite number of orbits", Comm. Algebra, 25(7), pp.2227~2236 (1997)
- Y. Hirano, "Rings with finitely many orbits under the regular action, Rings, modules, algebras, and abelian groups", pp.343~347 (2004), Dekker
- T. Y. Lam, "A First Course in Noncommutative Rings" (1991), Springer-Verlag
- W. K. Nicholson, "Introduction to Abstract Algebra" (1998), PWS