Dual coalgebras of algebras over commutative rings

Abuhlail, Jawad; Gómez-Torrecillas, José; Wisbauer, Robert

doi:10.1016/s0022-4049(99)00088-2

Cited by 18 publications

(33 citation statements)

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“…For some of our results, we assume that the ground ring is Artinian. Besides the new results, this paper extends results in [4] and "Kapitel 4" of my doctoral thesis [2]. A standard reference for the theory of linearly recursive sequences over rings and modules is the comprehensive work of Mikhalev et al [16].…”

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confidence: 64%

“…Linearly recursive (bi)sequences over arbitrary rings and modules were studied intensively by Nechaev et al (e.g.,[16,20,21]); however, the coalgebraic approach in their work was limited to the field case. Generalization to the case of arbitrary commutative ground rings was studied by several authors including Kurakin [12,13,14] and eventually Abuhlail, Gómez-Torrecillas, and Wisbauer [4].In this paper, we develop a coalgebraic aspect for the study of solutions of linear difference equations over arbitrary rings and modules. For some of our results, we assume that the ground ring is Artinian.…”

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confidence: 99%

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On linear difference equations over rings and modules

Abuhlail

2004

International Journal of Mathematics and Mathematical Sciences

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We develop a coalgebraic approach to the study of solutions of linear difference equations over modules and rings. Some known results about linearly recursive sequences over base fields are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary commutative ground rings.2000 Mathematics Subject Classification: 16W30, 39A99. Introduction.Although the theory of linear difference equations over base fields is well understood, the theory over arbitrary ground rings and modules is still under development. It is becoming more interesting and is gaining increasingly special importance mainly because of recent applications in coding theory and cryptography (e.g., [10,15]).In a series of papers, Taft et al. (e.g.,[17,22,24]) developed a coalgebraic aspect for the study of linearly recursive sequences over fields. Moreover, Grünenfelder et al. studied in [8,9] the linearly recursive sequences over finite-dimensional vector spaces. Linearly recursive (bi)sequences over arbitrary rings and modules were studied intensively by Nechaev et al. (e.g.,[16,20,21]); however, the coalgebraic approach in their work was limited to the field case. Generalization to the case of arbitrary commutative ground rings was studied by several authors including Kurakin [12,13,14] and eventually Abuhlail, Gómez-Torrecillas, and Wisbauer [4].In this paper, we develop a coalgebraic aspect for the study of solutions of linear difference equations over arbitrary rings and modules. For some of our results, we assume that the ground ring is Artinian. Besides the new results, this paper extends results in [4] and "Kapitel 4" of my doctoral thesis [2]. A standard reference for the theory of linearly recursive sequences over rings and modules is the comprehensive work of Mikhalev et al. [16]. For the theory of Hopf algebras, the reader may refer to any of the classical references (e.g., [1,19,23]).By R we denote a commutative ring with 1 R ≠ 0 R and with U(R) = {r ∈ R | r is invertible} the group of units of R. The category of R-(bi)modules will be denoted by ᏹ R . For an R-module M, we call an R-submodule K ⊂ M pure (in the sense of Cohn) if for every R-module N, the induced mapFor an R-algebra A and an A-module M, we call an A-submodule K ⊂ M R-cofinite if M/K is finitely generated in ᏹ R . For an R-algebra A, we denote by A the class of R-cofinite ideals. If A is an R-algebra with A a filter, then we define for every left

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On linear difference equations over rings and modules

Abuhlail

2004

International Journal of Mathematics and Mathematical Sciences

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“…Then I contains a reversible polynomial q x . By Lemma 4.12, R x x −1 / q x R x / q x which implies, by [1,Proposition 3.1], that R x x −1 / q x is finitely generated as an R-module. Therefore, R x x −1 /I is finitely generated as an R-module.…”

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confidence: 89%

“…By [1], there exists a monic polynomial f 1 x = a 0 + a 1 x + · · · + x n ∈ I ∩ R x . We know R x x −1 is a Hopf R-algebra with antipode…”

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Duality and Rational Modules in Hopf Algebras over Commutative Rings

2001

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“…([5, Remark 2.14, Proposition 2.15], [6]) Assume R to be Noetherian. Let A be an R-algebra and consider A * as an A-bimodule through the regular left and right actions…”

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confidence: 99%

Duality theorems for crossed products over rings

Abuhlail

2005

Journal of Algebra

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In this paper we improve and extend duality theorems for crossed products obtained by M. Koppinen (C. Chen) from the case of base fields (Dedekind domains) to the case of arbitrary Noetherian commutative ground rings under fairly weak conditions. In particular we extend an improved version of the celebrated Blattner-Montgomery duality theorem to the case of arbitrary Noetherian commutative ground rings.  2005 Elsevier Inc. All rights reserved.

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Dual coalgebras of algebras over commutative rings

Cited by 18 publications

References 7 publications

On linear difference equations over rings and modules

On linear difference equations over rings and modules

Duality and Rational Modules in Hopf Algebras over Commutative Rings

Duality theorems for crossed products over rings

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