A Note on Right Locally Finite Simple Ring Extensions
摘要
Thtoughout A will represent an (Artinian) simple ring, B a simple subring of A containing 1 of A, and V the centralizer of B in A. A ring extension A'/B' is said to be right locally finite if for any finite subset F' of A' the subring B'[F'] is right finite over B'.In [1], S. Takamatsu and the second author dealt with a right locally finite extension A/B such that V is simple and A=BN with the normalizer N of B in A, and proved that A/BV is right Iocally finite, which played an important role in the proof of Ll, Theorem].In this note, we shall prove the same without any restriction.Theorem.If A/B is right locally finite, then so is A/BV.Proof.Let F be an arbitrary finite subset of A, and choose an intermediate ring B' of A/B [F] such that AAB, is irreducible and the right rank [B':B]R is finite.Then by [2, Proposition 5.4 (b)] the centralizer V' of B' in A is a division ring and m=[V : V']R s{; [B' : B]R.Let {vi, v2, , vm} be a right V'-basis of V ancl set B" ==B[F, vi, , vm]==:..i b"tiB.Since every element of V' commutes with all the elements of B[F], we see that B'V'DV'B", namely, B"V' is a subrn ffiteOnfetlg o[(l4n;iilvl.BV)[F]=B"V'=Xj-rmib"j(BV)' which proves the right local