Summary: Radical Endomorphisms of Decomposable
Julius M. Zelmanowitz
University of California, Oakland, CA 94607 USA
An element of the Jacobson radical of the endomorphism ring of a decomposable
module is characterized in terms of its action on the components of the decompo-
sition. This extends to arbitrary decomposable modules a result previously known
only for the special case of free modules.
AMS Classification: 16N20, 16S50
The historical motivation for this study may be said to begin with a question
raised by N. Jacobson in "Structure of Rings", first published in 1956 .
On page 23 of that seminal text, Jacobson asked for a characterization of the
elements in the radical of the ring MI(R) of I × I row-finite matrices over an
arbitrary ring R. In general, it is not true that a matrix whose entries lie in
the radical of R is in the radical of MI (R), as will be demonstrated below.
For a ring R we let J(R) denote the Jacobson radical of R, and we consider
the free R-module of infinite rank consisting of all row vectors (r1, r2, . . .) such