An R-module M is called coretractable if Hom R (M/K , M) = 0 for any proper submodule K of M. In this paper we study coretractable modules and their endomorphism rings. It turns out that if all right R-modules are coretractable, then R is a right Kasch and (two-sided) perfect ring. However, the converse holds for commutative rings. Also, for a semi-injective coretractable module M R with S = End R (M), we show that u.dim( S S) = corank(M R ).2000 Mathematics subject classification: primary 16D10; secondary 16D40, 16L30.

O. IntroductionThroughout, R will denote a commutative ring with identity and every module is unitary. By B < A we mean that B is a proper submodule of A. For the submodules B and C of A, we let (B:C) = {r E R:rC C C_ B}.If P is a proper submodule of A such that ra E P, r E R, a E A implies either a E P or r E (P: A), then P is said to be prime in A. In case A = R, prime submodules coincide with prime ideals. The intersection of all prime submodules of A containing B is denoted by rad B. For an ideal I of R leta E A such that x = ra and rna E B for some n E N. We say that A satisfies the radical formula (A s.t.r.f.) if for every B < A, rad B = (E(B)). A ring R s.t.r.f, provided that if A is any R-module, then A s.t.r.f. 1. The radical of submodules McCasland and Moore in [5] showed that if R is a PID and A is a finitely generated R-module, then A s.t.r.f. Recently Jenkins and Smith in [2] have proved the following theorem. THEOREM 1.1. Let R be a Noetherian domain of finite global dimension. Then R is a Dedekind domain if and only if R s.t.r.f. They have also conjectures that a Noetherian domain R s.t.r.f, if and only if R has dimension 1. We shall provide a counter-example for the above conjecture. It is well-known that R = K[X 2, X3], where K is a field, is a Noetherian domain of Krull dimension 1 but not a Dedekind domain (see, for example, [6, Ex.6, p.99]). We shall show that K[X2,X 3] does not satisfy the radical formula. THEOREM 1.2. Let K be a field. Then R = K[X 2, X 3] does not satisfy the radical formula.

We give a characterization of algebraic functions over a field of positive characteristic and we then deduce that the Hadamard product of two algebraic series in several commutative variables over a field of positive characteristic is again algebraic.

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