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On a generalization of de Rham lemma

Kyoji Saito (1976)

Annales de l'institut Fourier

Let M be a free module over a noetherian ring. For ω 1 , ... , ω k M , let 𝒜 be the ideal generated by coefficients of ω 1 ... ω k . For an element ω p M with p < prof . 𝒜 , if ω ω 1 ... ω k = 0 , there exists η 1 , ... , η k p - 1 M such that ω = i = 1 k η i ω i .This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.

On a non-vanishing Ext

Laszlo Fuchs, Saharon Shelah (2003)

Rendiconti del Seminario Matematico della Università di Padova

On co-Gorenstein modules, minimal flat resolutions and dual Bass numbers

Zahra Heidarian, Hossein Zakeri (2015)

Colloquium Mathematicae

The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if R is a local U-ring and M is an Artinian R-module, then M is a co-Gorenstein R-module if and only if the complex H o m R ̂ ( ( , R ̂ ) , M ) is a minimal flat resolution for M when we choose a suitable triangular subset on R̂. Moreover we characterize the co-Gorenstein modules over a local U-ring and Cohen-Macaulay local U-ring.

On Cohen-Macaulay rings

Edgar E. Enochs, Jenda M. G. Overtoun (1994)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we use a characterization of R -modules N such that f d R N = p d R N to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting N to be the d t h local cohomology functor of R with respect to the maximal ideal where d is the Krull dimension of R .

On commutative rings whose maximal ideals are idempotent

Farid Kourki, Rachid Tribak (2019)

Commentationes Mathematicae Universitatis Carolinae

We prove that for a commutative ring R , every noetherian (artinian) R -module is quasi-injective if and only if every noetherian (artinian) R -module is quasi-projective if and only if the class of noetherian (artinian) R -modules is socle-fine if and only if the class of noetherian (artinian) R -modules is radical-fine if and only if every maximal ideal of R is idempotent.

On commutative rings whose prime ideals are direct sums of cyclics

M. Behboodi, A. Moradzadeh-Dehkordi (2012)

Archivum Mathematicum

In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring ( R , ) , the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R -modules; (2) = λ Λ R w λ where Λ is an index set and R / Ann ( w λ ) is a principal ideal ring for each λ Λ ; (3) Every prime ideal of R is a direct sum of at most...

On domains with ACC on invertible ideals

Stefania Gabelli (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

If A is a domain with the ascending chain condition on (integral) invertible ideals, then the group I ( A ) of its invertible ideals is generated by the set I m ( A ) of maximal invertible ideals. In this note we study some properties of I m ( A ) and we prove that, if I ( A ) is a free group on I m ( A ) , then A is a locally factorial Krull domain.

On endomorphisms of multiplication and comultiplication modules

H. Ansari-Toroghy, F. Farshadifar (2008)

Archivum Mathematicum

Let R be a ring with an identity (not necessarily commutative) and let M be a left R -module. This paper deals with multiplication and comultiplication left R -modules M having right End R ( M ) -module structures.

On finitely generated multiplication modules

R. Nekooei (2005)

Czechoslovak Mathematical Journal

We shall prove that if M is a finitely generated multiplication module and A n n ( M ) is a finitely generated ideal of R , then there exists a distributive lattice M ¯ such that S p e c ( M ) with Zariski topology is homeomorphic to S p e c ( M ¯ ) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R -modules M such that A n n ( M ) is a finitely generated ideal of R .

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