Modules which are invariant under idempotents of their envelopes

Le Van Thuyet; Phan Dan; Truong Cong Quynh

Displaying similar documents to “Modules which are invariant under idempotents of their envelopes”

Matrix rings with summand intersection property

F. Karabacak, Adnan Tercan (2003)

Czechoslovak Mathematical Journal

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A ring $R$ has right SIP (SSP) if the intersection (sum) of two direct summands of $R$ is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of $R$ by $M$ has SIP if and only if $R$ has SIP and $(1 - e) M e = 0$ for every idempotent $e$ in $R$ . Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

On commutative rings whose maximal ideals are idempotent

Farid Kourki, Rachid Tribak (2019)

Commentationes Mathematicae Universitatis Carolinae

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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.

Commutative rings whose certain modules decompose into direct sums of cyclic submodules

Farid Kourki, Rachid Tribak (2023)

Czechoslovak Mathematical Journal

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We provide some characterizations of rings $R$ for which every (finitely generated) module belonging to a class $𝒞$ of $R$ -modules is a direct sum of cyclic submodules. We focus on the cases, where the class $𝒞$ is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules.

Cellular covers of cotorsion-free modules

Rüdiger Göbel, José L. Rodríguez, Lutz Strüngmann (2012)

Fundamenta Mathematicae

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In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism $π ⁎ : H o m_{R} (G, G) ≅ H o m_{R} (G, H)$ , where π⁎(φ) = πφ for each $φ \in H o m_{R} (G, G)$ (where maps are acting on the left). On...

$\oplus$ -cofinitely supplemented modules

H. Çalışıcı, A. Pancar (2004)

Czechoslovak Mathematical Journal

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Let $R$ be a ring and $M$ a right $R$ -module. $M$ is called $\oplus$ -cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$ . In this paper various properties of the $\oplus$ -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus$ -cofinitely supplemented modules is $\oplus$ -cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$ -module is $\oplus$ -cofinitely supplemented. In addition, if $M$ has the...

Non-weight modules over the super Schrödinger algebra

Xinyue Wang, Liangyun Chen, Yao Ma (2024)

Czechoslovak Mathematical Journal

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We construct a family of non-weight modules which are free $U (𝔥)$ -modules of rank 2 over the $N = 1$ super Schrödinger algebra in $(1 + 1)$ -dimensional spacetime. We determine the isomorphism classes of these modules. In particular, free $U (𝔥)$ -modules of rank 2 over $𝔬𝔰𝔭 (1 | 2)$ are also constructed and classified. Moreover, we obtain the sufficient and necessary conditions for such modules to be simple.

Top-stable and layer-stable degenerations and hom-order

S. O. Smalø, A. Valenta (2007)

Colloquium Mathematicae

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Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration $M <_{d e g} N$ such that $^{t} M /^{t + 1} M ≃^{t} N /^{t + 1} N$ for all t. Given a module M with square-free top and a projective cover P, she showed that $d i m_{k} H o m (M, M) = d i m_{k} H o m (P, M)$ if and only if M has no proper degeneration $M <_{d e g} N$ where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from...

Gorenstein star modules and Gorenstein tilting modules

Peiyu Zhang (2021)

Czechoslovak Mathematical Journal

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We introduce the notion of Gorenstein star modules and obtain some properties and a characterization of them. We mainly give the relationship between $n$ -Gorenstein star modules and $n$ -Gorenstein tilting modules, see L. Yan, W. Li, B. Ouyang (2016), and a new characterization of $n$ -Gorenstein tilting modules.

Fundamental relation onm-idempotent hyperrings

Morteza Norouzi, Irina Cristea (2017)

Open Mathematics

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The γ*-relation defined on a general hyperring R is the smallest strongly regular relation such that the quotient R/γ* is a ring. In this note we consider a particular class of hyperrings, where we define a new equivalence, called [...] εm∗ $ε_{m}^{*}$ , smaller than γ* and we prove it is the smallest strongly regular relation on such hyperrings such that the quotient R/ [...] εm∗ $ε_{m}^{*}$ is a ring. Moreover, we introduce the concept of m-idempotent hyperrings, show that they are a characterization for...

Some results on the cofiniteness of local cohomology modules

Sohrab Sohrabi Laleh, Mir Yousef Sadeghi, Mahdi Hanifi Mostaghim (2012)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$ , $M$ an $R$ -module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $𝒜ℱ$ modules. The main result is that if the $R$ -module ${Ext}_{R}^{t} (R / 𝔞, M)$ is finite (finitely generated), $H_{𝔞}^{i} (M)$ is $𝔞$ -cofinite for all $i < t$ and $H_{𝔞}^{t} (M)$ is minimax then $H_{𝔞}^{t} (M)$ is $𝔞$ -cofinite. As a consequence we show that if $M$ and $N$ are finite $R$ -modules and $H_{𝔞}^{i} (N)$ is minimax for all $i < t$ then the set of associated prime ideals of the generalized local cohomology...