The Jacobson Radical of Some Endomorphism Rings

Abstract

In what follows, G always denotes an abelian group where OG is its zero and EG is its endomorphism ring. If S is a ring, then JS is to be its Jacobson radical, Os is to be its zero, while its unity, if any, is denoted by 1s. In ansver to a query of Jacobson’s [3, p. 23], Patterson [5] [4] showed that the Jacobson radical of the ring of row-finite matrices over S is the ring of row-finite matrices over JS if and only if JS has a right-vanishing condition due to Levitzki, namely that if b0,b1,... is any sequence chosen from JS, then there is a least non-negative integer n (depending upon the sequence) for which the product bO...bn (just bO if n = O) vanishes. If n depends only upon the initial sequence member b, then [4] this sort of right vanishing is called uniform.