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The Jacobson Radical of Some Endomorphism Rings

In: Études sur les Groupes abéliens / Studies on Abelian Groups

Author

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  • Franklin Haimo

    (Washington University)

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.

Suggested Citation

  • Franklin Haimo, 1968. "The Jacobson Radical of Some Endomorphism Rings," Springer Books, in: Études sur les Groupes abéliens / Studies on Abelian Groups, chapter 0, pages 143-146, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-46146-0_9
    DOI: 10.1007/978-3-642-46146-0_9
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