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Relative theory in subcategories

Soud Khalifa Mohamed (2009)

Colloquium Mathematicae

We generalize the relative (co)tilting theory of Auslander-Solberg in the category mod Λ of finitely generated left modules over an artin algebra Λ to certain subcategories of mod Λ. We then use the theory (relative (co)tilting theory in subcategories) to generalize one of the main result of Marcos et al. [Comm. Algebra 33 (2005)].

Relative weak derived functors

Panneerselvam Prabakaran (2020)

Commentationes Mathematicae Universitatis Carolinae

Let R be a ring, n a fixed non-negative integer, 𝒲 ℐ the class of all left R -modules with weak injective dimension at most n , and 𝒲 ℱ the class of all right R -modules with weak flat dimension at most n . Using left (right) 𝒲 ℐ -resolutions and the left derived functors of Hom we study the weak injective dimensions of modules and rings. Also we prove that - ⊗ - is right balanced on ℳ R × R ℳ by 𝒲 ℱ × 𝒲 ℐ , and investigate the global right 𝒲 ℐ -dimension of R ℳ by right derived functors of ⊗ .

Relatively coarse sequential convergence

Roman Frič, Fabio Zanolin (1997)

Czechoslovak Mathematical Journal

We generalize the notion of a coarse sequential convergence compatible with an algebraic structure to a coarse one in a given class of convergences. In particular, we investigate coarseness in the class of all compatible convergences (with unique limits) the restriction of which to a given subset is fixed. We characterize such convergences and study relative coarseness in connection with extensions and completions of groups and rings. E.g., we show that: (i) each relatively coarse dense group precompletion...

Relatively exact modules

Ladislav Bican (2003)

Commentationes Mathematicae Universitatis Carolinae

Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice ℰ τ ( M ) of submodules of a torsionfree module M consisting of all submodules N of M such that M / N is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of M / N is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application...

Remarks on Sekine quantum groups

Jialei Chen, Shilin Yang (2022)

Czechoslovak Mathematical Journal

We first describe the Sekine quantum groups 𝒜 k (the finite-dimensional Kac algebra of Kac-Paljutkin type) by generators and relations explicitly, which maybe convenient for further study. Then we classify all irreducible representations of 𝒜 k and describe their representation rings r ( 𝒜 k ) . Finally, we compute the the Frobenius-Perron dimension of the Casimir element and the Casimir number of r ( 𝒜 k ) .

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