Testing flatness and computing rank of a module using syzygies

Oswaldo Lezama

Displaying similar documents to “Testing flatness and computing rank of a module using syzygies”

Some results on $G_{C}$ -flat dimension of modules

Ramalingam Udhayakumar, Intan Muchtadi-Alamsyah, Chelliah Selvaraj (2019)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we study some properties of $G_{C}$ -flat $R$ -modules, where $C$ is a semidualizing module over a commutative ring $R$ and we investigate the relation between the $G_{C}$ -yoke with the $C$ -yoke of a module as well as the relation between the $G_{C}$ -flat resolution and the flat resolution of a module over $G F$ -closed rings. We also obtain a criterion for computing the $G_{C}$ -flat dimension of modules.

Relative Gorenstein injective covers with respect to a semidualizing module

Elham Tavasoli, Maryam Salimi (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$ -module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_{C}$ -injective module $G$ , the character module $G^{+}$ is $G_{C}$ -flat, then the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

Finitistic dimension and restricted injective dimension

Dejun Wu (2015)

Czechoslovak Mathematical Journal

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We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$ -module with $A = {End}_{R} T$ . If $_{R} T$ is selforthogonal, then we show that $rid (T_{A}) \leq findim (A_{A}) \leq findim (_{R} T) + rid (T_{A})$ . Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left $R$ -module with finite injective dimension, then $rid (T_{A}) \leq findim (A_{A}) \leq fin . inj . \dim (_{R} R) + rid (T_{A})$ . Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.

Special modules for $R (PSL (2, q))$

Liufeng Cao, Huixiang Chen (2023)

Czechoslovak Mathematical Journal

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Let $R$ be a fusion ring and $R_{ℂ} : = R \otimes_{ℤ} ℂ$ be the corresponding fusion algebra. We first show that the algebra $R_{ℂ}$ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, $R_{ℂ}$ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra $R (PSL (2, q)) : = r (PSL (2, q)) \otimes_{ℤ} ℂ$ up to isomorphism, where $r (PSL (2, q))$ is the...

On soluble groups of module automorphisms of finite rank

Bertram A. F. Wehrfritz (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring, $M$ an $R$ -module and $G$ a group of $R$ -automorphisms of $M$ , usually with some sort of rank restriction on $G$ . We study the transfer of hypotheses between $M / C_{M} (G)$ and $[M, G]$ such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose $[M, G]$ is $R$ -Noetherian. If $G$ has finite rank, then $M / C_{M} (G)$ also is $R$ -Noetherian. Further, if $[M, G]$ is $R$ -Noetherian and if only certain abelian...

Some results on top local cohomology modules with respect to a pair of ideals

Saeed Jahandoust, Reza Naghipour (2020)

Mathematica Bohemica

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Let $I$ and $J$ be ideals of a Noetherian local ring $(R, 𝔪)$ and let $M$ be a nonzero finitely generated $R$ -module. We study the relation between the vanishing of $H_{I, J}^{dim M} (M)$ and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian $R$ -module $M / J M$ is equal to its integral closure relative to the Artinian $R$ -module $H_{I, J}^{dim M} (M)$ .

Ding projective and Ding injective modules over trivial ring extensions

Lixin Mao (2023)

Czechoslovak Mathematical Journal

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Let $R ⋉ M$ be a trivial extension of a ring $R$ by an $R$ - $R$ -bimodule $M$ such that $M_{R}$ , $_{R} M$ , ${(R, 0)}_{R ⋉ M}$ and $_{R ⋉ M} (R, 0)$ have finite flat dimensions. We prove that $(X, α)$ is a Ding projective left $R ⋉ M$ -module if and only if the sequence $M \otimes_{R} M \otimes_{R} X \overset{M \otimes α}{⟶} M \otimes_{R} X \overset{α}{\to} X$ is exact and $coker (α)$ is a Ding projective left $R$ -module. Analogously, we explicitly describe Ding injective $R ⋉ M$ -modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.

On $n$ -submodules and $G . n$ -submodules

Somayeh Karimzadeh, Javad Moghaderi (2023)

Czechoslovak Mathematical Journal

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We investigate some properties of $n$ -submodules. More precisely, we find a necessary and sufficient condition for every proper submodule of a module to be an $n$ -submodule. Also, we show that if $M$ is a finitely generated $R$ -module and $\sqrt{{Ann}_{R} (M)}$ is a prime ideal of $R$ , then $M$ has $n$ -submodule. Moreover, we define the notion of $G . n$ -submodule, which is a generalization of the notion of $n$ -submodule. We find some characterizations of $G . n$ -submodules and we examine the way the aforementioned notions are related...

Coherence relative to a weak torsion class

Zhanmin Zhu (2018)

Czechoslovak Mathematical Journal

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Let $R$ be a ring. A subclass $𝒯$ of left $R$ -modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $𝒯$ be a weak torsion class of left $R$ -modules and $n$ a positive integer. Then a left $R$ -module $M$ is called $𝒯$ -finitely generated if there exists a finitely generated submodule $N$ such that $M / N \in 𝒯$ ; a left $R$ -module $A$ is called $(𝒯, n)$ -presented if there exists an exact sequence of left $R$ -modules $0 ⟶ K_{n - 1} ⟶ F_{n - 1} ⟶ \dots ⟶ F_{1} ⟶ F_{0} ⟶ M ⟶ 0$ such that $F_{0}, \dots, F_{n - 1}$ are finitely generated free and $K_{n - 1}$ is $𝒯$ -finitely generated;...

Some homological properties of amalgamated modules along an ideal

Hanieh Shoar, Maryam Salimi, Abolfazl Tehranian, Hamid Rasouli, Elham Tavasoli (2023)

Czechoslovak Mathematical Journal

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Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$ , $f : R \to S$ be a ring homomorphism, $M$ be an $R$ -module, $N$ be an $S$ -module, and let $ϕ : M \to N$ be an $R$ -homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R ⋈^{f} J$ was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of $(R ⋈^{f} J)$ -module called the amalgamation of $M$ and $N$ along $J$ with respect to $ϕ$ , and denoted by $M ⋈^{ϕ} J N$ . We study some homological properties of the $(R ⋈^{f} J)$ -module $M ⋈^{ϕ} J N$ . Among...

On generalized CS-modules

Qingyi Zeng (2015)

Czechoslovak Mathematical Journal

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An $𝒮$ -closed submodule of a module $M$ is a submodule $N$ for which $M / N$ is nonsingular. A module $M$ is called a generalized CS-module (or briefly, GCS-module) if any $𝒮$ -closed submodule $N$ of $M$ is a direct summand of $M$ . Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right $R$ -modules are projective if and only if all right $R$ -modules are GCS-modules.

Derived dimension via $τ$ -tilting theory

Yingying Zhang (2021)

Czechoslovak Mathematical Journal

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Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support $τ$ -tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given $τ$ -tilting module.

Generalized tilting modules over ring extension

Zhen Zhang (2019)

Czechoslovak Mathematical Journal

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Let $Γ$ be a ring extension of $R$ . We show the left $Γ$ -module $U = Γ \otimes_{R} C$ with the endmorphism ring End $_{Γ} U = Δ$ is a generalized tilting module when $_{R} C$ is a generalized tilting module under some conditions.