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Ina recently published paper [3 ], the elements of the Jacobson radical of a ring of row-finite matrices over an arbitrary ring R are characterized as those matrices with entries in the Jacobson radical of R which have a vanishing set of column ideals. In this paper, the characterization is extended to include the endomorphism ring of an arbitrary projective module. In the process we offer a greatly simplified proof of the theorem for row-finite matrices. Throughout A will be an associative ring, and A an infinite index set which we will assume to be well-ordered by the ordinals, A = {1,2, ••• }. For a ring A, we will let A' denote the ring obtained from A by adjoining an identity element in the customary manner, and /(A) will denote the Jacobson radical of A. Recall that for A a ring with 1, a left A-module P is projective ii given any homomorphism f:P—>N and any surjection g:M—>N of left A-modules there exists a homomorphism h:P—>M with f=h o g. It is well known [2, p. 86] that P is projective if and only if there exist elements |xx[XGA}çP and homomorphisms {/x|X€EA} ÇHomfi(P, A) with x = X)xsa (x/\)xx for each x£P (the sum having but finitely many nonzero terms). We will call such a family {(xx, /x) | X£A} a projective coordinate system for P. We need one more definition before we can state our main theorem. A set of left ideals {.4x| X£A J of a ring A is called a vanishing set of left ideals if given any sequence ai, at, ■ ■ ■ with a<G^4x, for distinct Xi in A, there exists an integer « for which aia2 ■ ■ • a„ = 0. Theorem 1. Let P be a projective left R-module, R a ring with 1, and let <pEE = HomRiP, P), endomorphisms acting on the right. Then the following conditions are equivalent. ii) d>EJiE). (ii) There exists an infinite projective coordinate system { (xx, /x) | X Received by the editors December 12, 1969. AMS 1969 subject classifications. Primary 1630, 1648; Secondary 1640.

The Jacobson radical of the endomorphism ring of a projective module.