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A generalization of the finiteness problem of the local cohomology modules

Ahmad Abbasi, Hajar Roshan-Shekalgourabi (2014)

Czechoslovak Mathematical Journal

Let R be a commutative Noetherian ring and 𝔞 an ideal of R . We introduce the concept of 𝔞 -weakly Laskerian R -modules, and we show that if M is an 𝔞 -weakly Laskerian R -module and s is a non-negative integer such that Ext R j ( R / 𝔞 , H 𝔞 i ( M ) ) is 𝔞 -weakly Laskerian for all i < s and all j , then for any 𝔞 -weakly Laskerian submodule X of H 𝔞 s ( M ) , the R -module Hom R ( R / 𝔞 , H 𝔞 s ( M ) / X ) is 𝔞 -weakly Laskerian. In particular, the set of associated primes of H 𝔞 s ( M ) / X is finite. As a consequence, it follows that if M is a finitely generated R -module and N is an 𝔞 -weakly...

A simple proof of uniqueness for torsion modules over principal ideal domains.

J. L. García Roig (1985)

Stochastica

The aim of this note is to give an alternative proof of uniqueness for the decomposition of a finitely generated torsion module over a P.I.D. (= principal ideal domain) as a direct sum of indecomposable submodules.Our proof tries to mimic as far as we can the standard procedures used when dealing with vector spaces.For the sake of completeness we also include a proof of the existence theorem.

Almost free splitters

Rüdiger Göbel, Saharon Shelah (1999)

Colloquium Mathematicae

Let R be a subring of the rationals. We want to investigate self splitting R-modules G, that is, such that E x t R ( G , G ) = 0 . For simplicity we will call such modules splitters (see [10]). Also other names like stones are used (see a dictionary in Ringel’s paper [8]). Our investigation continues [5]. In [5] we answered an open problem by constructing a large class of splitters. Classical splitters are free modules and torsion-free, algebraically compact ones. In [5] we concentrated on splitters which are larger...

An addendum and corrigendum to "Almost free splitters" (Colloq. Math. 81 (1999), 193-221)

Rüdiger Göbel, Saharon Shelah (2001)

Colloquium Mathematicae

Let R be a subring of the rational numbers ℚ. We recall from [3] that an R-module G is a splitter if E x t ¹ R ( G , G ) = 0 . In this note we correct the statement of Main Theorem 1.5 in [3] and discuss the existence of non-free splitters of cardinality ℵ₁ under the negation of the special continuum hypothesis CH.

An algebraic framework for linear identification

Michel Fliess, Hebertt Sira-Ramírez (2003)

ESAIM: Control, Optimisation and Calculus of Variations

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.

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