Rings whose finitely generated modules are direct sums of cyclics

Shores, Thomas S.; Wiegand, Roger

doi:10.1016/0021-8693(74)90178-1

Cited by 38 publications

(21 citation statements)

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“…We follow Shores and Wiegand (1974a), by defining a canonical form for an R-module E to be a decomposition E ' R=I 1 È R=I 2 È Á Á Á È R=I n , where I 1 I 2 Á Á Á I n 6 ¼ R, and by calling a ring R a CF-ring if every direct sum of finitely many cyclic modules has a canonical form.…”

Section: K-dimension Of Commutative Arithmetic Rings 3151mentioning

confidence: 99%

The λ-Dimension of Commutative Arithmetic Rings

Couchot

2003

Communications in Algebra

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It is shown that every commutative arithmetic ring R has l-dimension3. An example of a commutative Kaplansky ring with l-dimension 3 is given. Moreover, if R satisfies one of the following conditions, semi-local, semi-prime, self fp-injective, zero-Krull dimensional, CF or FSI then l-dim(R) 2. It is also shown that every zero-Krull dimensional commutative arithmetic ring is a Kaplansky ring and an adequate ring, that every Bézout ring with compact minimal prime spectrum is Hermite and that each Bézout fractionnally self fp-injective ring is a Kaplansky ring.

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Section: K-dimension Of Commutative Arithmetic Rings 3151mentioning

confidence: 99%

The λ-Dimension of Commutative Arithmetic Rings

Couchot

2003

Communications in Algebra

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“…A ring R is called a CF-ring if every R-module M which is a direct sum of finitely many cyclic R-modules has a canonical form decomposition, i.e. [15]). For the definitions of the other types of rings used in the following two results, we refer the reader to [7].…”

Section: Proposition 25 a Free R-module F Is Weakly In If And Only If...mentioning

confidence: 99%

On Two Classes of Modules Related to CS Trivial Extensions

Kourki¹,

Tribak²

2021

Preprint

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“…A ring is an FGC ring if every finitely generated module over the ring is a direct sum of cyclic submodules ( [9]). It was noted by Hadwin and Kerr in [3, p. 3] that a finite direct sum of cyclic modules is reflexive.…”

Section: Definitions and Notationmentioning

confidence: 99%

Two questions on scalar-reflexive rings

Snashall¹

1992

Proc. Amer. Math. Soc.

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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.

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Rings whose finitely generated modules are direct sums of cyclics

Cited by 38 publications

References 16 publications

The λ-Dimension of Commutative Arithmetic Rings

The λ-Dimension of Commutative Arithmetic Rings

On Two Classes of Modules Related to CS Trivial Extensions

Two questions on scalar-reflexive rings

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