Two questions on scalar-reflexive rings

Abstract

Abstract.A module M over a commutative ring R with unity is reflexive if the only Ä-endomorphisms of M leaving invariant every submodule of M are the scalar multiplications by elements of R . A commutative ring R is scalarreflexive if every finitely generated .R-module is reflexive. A local version of scalar-reflexivity is introduced, and it is shown that every locally scalar-reflexive ring is scalar-reflexive. An example is given of a scalar-reflexive domain that is not /¡-local. This answers a question posed by Hadwin and Kerr. Theorem 7 gives eight equivalent conditions on an A-local domain for it to be scalarreflexive, thus classifying the scalar-reflexive /¡-local domains.A module M over a commutative ring R with unity is said to be reflexive if the only /?-endomorphisms of M leaving invariant every submodule of M are the left scalar multiplications by elements of R. In [4], Hadwin and Kerr defined a commutative ring R to be scalar-reflexive if every finitely generated Ä-module is reflexive. Throughout this paper all rings are commutative with unity. This paper considers two questions of Hadwin and Kerr on scalarreflexive rings.Hadwin and Kerr ask in [3, p. 7] whether the property of being scalar-reflexive is preserved under localisations. They characterised the local scalar-reflexive rings in [4, Theorem 6], showing that a local ring is scalar-reflexive if and only if it is an almost maximal valuation ring (see Proposition 1). This result of Hadwin and Kerr motivates Definition 2, where a ring is defined to be locally scalar-reflexive if every localisation at a maximal ideal is scalar-reflexive. The first theorem in this paper proves that every locally scalar-reflexive ring is scalarreflexive. This result is given in Theorem 4 and provides a converse to the question raised by Hadwin and Kerr. A second question of Hadwin and Kerr concerns the characterisation of the scalar-reflexive domains and they ask in [4, p. 318] whether every scalar-reflexive domain is an /¡-local domain. Matlis defined an /¡-local domain in [5, §8] to be a domain such that every nonzero prime ideal is contained in a unique maximal ideal, and every nonzero element is contained in only finitely many maximal ideals. Example 5 provides an example of a scalar-reflexive domain that is not /¡-local, thus answering this second question in the negative.