Finite groups of OTP projective representation type over a complete discrete valuation domain of positive characteristic

Leonid F. Barannyk; Dariusz Klein

Displaying similar documents to “Finite groups of OTP projective representation type over a complete discrete valuation domain of positive characteristic”

Finite groups of OTP projective representation type

Leonid F. Barannyk (2012)

Colloquium Mathematicae

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Let K be a field of characteristic p > 0, K* the multiplicative group of K and $G = G_{p} \times B$ a finite group, where $G_{p}$ is a p-group and B is a p’-group. Denote by $K^{λ} G$ a twisted group algebra of G over K with a 2-cocycle λ ∈ Z²(G,K*). We give necessary and sufficient conditions for G to be of OTP projective K-representation type, in the sense that there exists a cocycle λ ∈ Z²(G,K*) such that every indecomposable $K^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $K^{λ} G_{p}$ -module...

On twisted group algebras of OTP representation type

Leonid F. Barannyk, Dariusz Klein (2012)

Colloquium Mathematicae

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Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G = G_{p} \times B$ is a finite group, where $G_{p}$ is a p-group and B is a p’-group. Denote by $S^{λ} G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{λ} G$ to be of OTP representation type, in the sense that every indecomposable $S^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $S^{λ} G_{p}$ -module V and an irreducible...

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 twisted group algebras of OTP representation type over the ring of p-adic integers

Leonid F. Barannyk, Dariusz Klein (2016)

Colloquium Mathematicae

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Let $ℤ ̂_{p}$ be the ring of p-adic integers, $U (ℤ ̂_{p})$ the unit group of $ℤ ̂_{p}$ and $G = G_{p} \times B$ a finite group, where $G_{p}$ is a p-group and B is a p’-group. Denote by $ℤ ̂_{p}^{λ} G$ the twisted group algebra of G over $ℤ ̂_{p}$ with a 2-cocycle $λ \in Z ² (G, U (ℤ ̂_{p}))$ . We give necessary and sufficient conditions for $ℤ ̂_{p}^{λ} G$ to be of OTP representation type, in the sense that every indecomposable $ℤ ̂_{p}^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $ℤ ̂_{p}^{λ} G_{p}$ -module V and an irreducible $ℤ ̂_{p}^{λ} B$ -module W.

Rings whose nonsingular right modules are $R$ -projective

Yusuf Alagöz, Sinem Benli, Engin Büyükaşık (2021)

Commentationes Mathematicae Universitatis Carolinae

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A right $R$ -module $M$ is called $R$ -projective provided that it is projective relative to the right $R$ -module $R_{R}$ . This paper deals with the rings whose all nonsingular right modules are $R$ -projective. For a right nonsingular ring $R$ , we prove that $R_{R}$ is of finite Goldie rank and all nonsingular right $R$ -modules are $R$ -projective if and only if $R$ is right finitely $Σ$ - $C S$ and flat right $R$ -modules are $R$ -projective. Then, $R$ -projectivity of the class of nonsingular injective right modules is also considered....

Relative tilting modules with respect to a semidualizing module

Maryam Salimi (2019)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, and let $C$ be a semidualizing $R$ -module. The notion of $C$ -tilting $R$ -modules is introduced as the relative setting of the notion of tilting $R$ -modules with respect to $C$ . Some properties of tilting and $C$ -tilting modules and the relations between them are mentioned. It is shown that every finitely generated $C$ -tilting $R$ -module is $C$ -projective. Finally, we investigate some kernel subcategories related to $C$ -tilting modules.

Twisted group rings of strongly unbounded representation type

Leonid F. Barannyk, Dariusz Klein (2004)

Colloquium Mathematicae

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Let S be a commutative local ring of characteristic p, which is not a field, S* the multiplicative group of S, W a subgroup of S*, G a finite p-group, and $S^{λ} G$ a twisted group ring of the group G and of the ring S with a 2-cocycle λ ∈ Z²(G,S*). Denote by $I n d_{m} (S^{λ} G)$ the set of isomorphism classes of indecomposable $S^{λ} G$ -modules of S-rank m. We exhibit rings $S^{λ} G$ for which there exists a function $f_{λ} : ℕ \to ℕ$ such that $f_{λ} (n) \geq n$ and $I n d_{f_{λ} (n)} (S^{λ} G)$ is an infinite set for every natural n > 1. In special cases $f_{λ} (ℕ)$ contains every natural...

Homological dimensions for endomorphism algebras of Gorenstein projective modules

Aiping Zhang, Xueping Lei (2024)

Czechoslovak Mathematical Journal

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Let $A$ be a CM-finite Artin algebra with a Gorenstein-Auslander generator $E$ , $M$ be a Gorenstein projective $A$ -module and $B = {End}_{A} M$ . We give an upper bound for the finitistic dimension of $B$ in terms of homological data of $M$ . Furthermore, if $A$ is $n$ -Gorenstein for $2 \leq n < \infty$ , then we show the global dimension of $B$ is less than or equal to $n$ plus the $B$ -projective dimension of ${Hom}_{A} (M, E) .$ As an application, the global dimension of ${End}_{A} E$ is less than or equal to $n$ .

Representations of a class of positively based algebras

Shiyu Lin, Shilin Yang (2023)

Czechoslovak Mathematical Journal

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We investigate the representation theory of the positively based algebra $A_{m, d}$ , which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that $A_{m, d}$ is of finite representative type if $d \leq 4$ , of tame type if $d = 5$ , and of wild type if $d \geq 6 .$ In the case when $d \leq 4$ , all indecomposable representations of $A_{m, d}$ are constructed. Furthermore, their right cell representations as well as left cell representations of $A_{m, d}$ are described. ...

$α$ -modules and generalized submodules

Rafiquddin Rafiquddin, Ayazul Hasan, Mohammad Fareed Ahmad (2019)

Communications in Mathematics

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A QTAG-module $M$ is an $α$ -module, where $α$ is a limit ordinal, if $M / H_{β} (M)$ is totally projective for every ordinal $β < α$ . In the present paper $α$ -modules are studied with the help of $α$ -pure submodules, $α$ -basic submodules, and $α$ -large submodules. It is found that an $α$ -closed $α$ -module is an $α$ -injective. For any ordinal $ω \leq α \leq ω_{1}$ we prove that an $α$ -large submodule $L$ of an $ω_{1}$ -module $M$ is summable if and only if $M$ is summable.

Some isomorphic properties in projective tensor products

Ioana Ghenciu (2022)

Commentationes Mathematicae Universitatis Carolinae

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We give sufficient conditions implying that the projective tensor product of two Banach spaces $X$ and $Y$ has the $p$ -sequentially Right and the $p$ - $L$ -limited properties, $1 \leq p < \infty$ .

Characterizing projective general unitary groups ${PGU}_{3} (q^{2})$ by their complex group algebras

Farrokh Shirjian, Ali Iranmanesh (2017)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. Let $X_{1} (G)$ be the first column of the ordinary character table of $G$ . We will show that if $X_{1} (G) = X_{1} ({PGU}_{3} (q^{2}))$ , then $G ≅ {PGU}_{3} (q^{2})$ . As a consequence, we show that the projective general unitary groups ${PGU}_{3} (q^{2})$ are uniquely determined by the structure of their complex group algebras.

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...

A note on generalizations of semisimple modules

Engin Kaynar, Burcu N. Türkmen, Ergül Türkmen (2019)

Commentationes Mathematicae Universitatis Carolinae

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A left module $M$ over an arbitrary ring is called an $ℛ𝒟$ -module (or an $ℛ𝒮$ -module) if every submodule $N$ of $M$ with $Rad (M) \subseteq N$ is a direct summand of (a supplement in, respectively) $M$ . In this paper, we investigate the various properties of $ℛ𝒟$ -modules and $ℛ𝒮$ -modules. We prove that $M$ is an $ℛ𝒟$ -module if and only if $M = Rad (M) \oplus X$ , where $X$ is semisimple. We show that a finitely generated $ℛ𝒮$ -module is semisimple. This gives us the characterization of semisimple rings in terms of $ℛ𝒮$ -modules. We completely determine the structure...

The module of vector-valued modular forms is Cohen-Macaulay

Richard Gottesman (2020)

Czechoslovak Mathematical Journal

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Let $H$ denote a finite index subgroup of the modular group $Γ$ and let $ρ$ denote a finite-dimensional complex representation of $H .$ Let $M (ρ)$ denote the collection of holomorphic vector-valued modular forms for $ρ$ and let $M (H)$ denote the collection of modular forms on $H$ . Then $M (ρ)$ is a $ℤ$ -graded $M (H)$ -module. It has been proven that $M (ρ)$ may not be projective as a $M (H)$ -module. We prove that $M (ρ)$ is Cohen-Macaulay as a $M (H)$ -module. We also explain how to apply this result to prove that if $M (H)$ is a polynomial ring, then...

Strongly $(𝒯, n)$ -coherent rings, $(𝒯, n)$ -semihereditary rings and $(𝒯, n)$ -regular rings

Zhanmin Zhu (2020)

Czechoslovak Mathematical Journal

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Let $𝒯$ be a weak torsion class of left $R$ -modules and $n$ a positive integer. A left $R$ -module $M$ is called $(𝒯, n)$ -injective if ${Ext}_{R}^{n} (C, M) = 0$ for each $(𝒯, n + 1)$ -presented left $R$ -module $C$ ; a right $R$ -module $M$ is called $(𝒯, n)$ -flat if ${Tor}_{n}^{R} (M, C) = 0$ for each $(𝒯, n + 1)$ -presented left $R$ -module $C$ ; a left $R$ -module $M$ is called $(𝒯, n)$ -projective if ${Ext}_{R}^{n} (M, N) = 0$ for each $(𝒯, n)$ -injective left $R$ -module $N$ ; the ring $R$ is called strongly $(𝒯, n)$ -coherent if whenever $0 \to K \to P \to C \to 0$ is exact, where $C$ is $(𝒯, n + 1)$ -presented and $P$ is finitely generated projective, then $K$ is $(𝒯, n)$ -projective; the ring $R$ is called...

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 results on $(n, d)$ -injective modules, $(n, d)$ -flat modules and $n$ -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $n, d$ be two non-negative integers. A left $R$ -module $M$ is called $(n, d)$ -injective, if ${Ext}^{d + 1} (N, M) = 0$ for every $n$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called $(n, d)$ -flat, if ${Tor}_{d + 1} (V, N) = 0$ for every $n$ -presented left $R$ -module $N$ . A left $R$ -module $M$ is called weakly $n$ - $F P$ -injective, if ${Ext}^{n} (N, M) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called weakly $n$ -flat, if ${Tor}_{n} (V, N) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . In this paper, we give some characterizations and properties of $(n, d)$ -injective modules and $(n, d)$ -flat modules in...