Abstract
This paper is a continuation of the work in [S]. Throughout? R is a commutative ring with identity and all modules are assumed to be unitary, An R-endomorphism T on an R-module JZ%! is locally scalar if, for each x in J%?. there is an r.r in R such that Tx= r,x. Equivalently, T is locally scalar on JZ if T(,I’) c ,V for every R-submodule ,I,‘” of ,&‘. An R-module JC since we have completely characterized strongly scalar-reflexive rings (Theorem 5), we feel that the new terminology is more appropriate. The notion of scalar-reflexivity was motivated by results on algebraic reflexivity [4] that were used to prove reflexivity theorems for Banachspace operators [4, 61. In this paper we show that a ring is strongly scalar-reflexive if and only if it is a finite direct sum of maximal valuation rings. (Note that we do not require valuation rings to be domains.) We also show that a local ring is scalar-reflexive if and only if it is an almost maximal valuation ring. In addition, we show that every Dedekind domain is scalar-reflexive, and we 311 0021~8693189 163.W
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