Journal of Algebra and Its Applications最新文献

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A note on odd periodic derived Hall algebras 关于奇周期派生霍尔代数的说明

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-06-06 DOI: 10.1142/s0219498825502822

Haicheng Zhang, Xinran Zhang, Zhiwei Zhu

Let $m$ be an odd positive integer and $D_{m} (𝒜)$ be the $m$ -periodic derived category of a finitary hereditary Abelian category $𝒜$ . In this note, we prove that there is an embedding of algebras from the derived Hall algebra of $D_{m} (𝒜)$ defined by Xu–Chen [Hall algebras of odd periodic triangulated categories, Algebr. Represent. Theory16(3) (2013) 673–687] to the extended derived Hall algebra of $D_{m} (𝒜)$ defined in [H. Zhang, Periodic derived Hall algebras of hereditary Abelian categories, preprint (2023), arXiv:2303.02912v2]. This homomorphism is given on basis elements, rather than just on generating elements.

设 m 为奇数正整数，Dm(𝒜) 为有限遗传阿贝尔范畴 𝒜 的 m 周期派生范畴。在本注释中，我们将证明存在一个由许琛定义的 Dm(𝒜) 的派生霍尔代数的代数嵌入[Hall algebras of odd periodic triangulated categories, Algebr.Represent.Theory16(3) (2013) 673-687]中定义的 Dm(𝒜)的扩展导出霍尔代数[H. Zhang, Periodic derived Hall algege of Dm(𝒜) defined in [H.Zhang, Periodic derived Hall algebras of hereditary Abelian categories, preprint (2023), arXiv:2303.02912v2] 中定义的 Dm(𝒜) 的扩展导出霍尔代数。这个同态是在基元上给出的，而不仅仅是在生成元上。

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引用次数: 0

Abelian groups whose endomorphism rings are V-rings 内态环为 V 环的阿贝尔群

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-06-04 DOI: 10.1142/s0219498825502871

Afshin Amini, Babak Amini, Ehsan Momtahan

We study Abelian groups whose endomorphism rings are V-rings. Let $G$ be a non-reduced Abelian group, We prove that $End (G)$ is a V-ring on either side if and only if $G = B \oplus ℚ^{n}$ where $B$ is a tame elementary Abelian group. We observe that a reduced group whose endomorphism is a V-ring, is an sp-group. Recognizing that $End (G)$ is also an sp-group of $\prod_{p \in ℙ} End (G_{p})$ , we show that $End (G) / \oplus End (G_{p})$ is a V-ring if and only if $End (G)$ is a V-ring.

我们研究的是其内定环是 V 环的无边群。让 G 是一个非还原的阿贝尔群，我们证明，当且仅当 G=B⊕ℚn 时，End(G) 的任一边都是一个 V 环，其中 B 是一个驯服的基本阿贝尔群。我们注意到，一个还原群的内形是一个 V 环，它是一个 sp 群。认识到 End(G) 也是∏p∈ℙEnd(Gp) 的一个 sp 群，我们证明当且仅当 End(G) 是一个 V 环时，End(G)/⊕End(Gp) 是一个 V 环。

引用次数: 0

Künneth formulas for Cotor 科托尔的库奈特公式

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-06-04 DOI: 10.1142/s0219498825502652

A. Salch

We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e. Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the $0$ th Cotor group, i.e. the cotensor product. We give topological applications in the form of consequences for the $E_{2}$ -term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.

我们研究的问题是如何计算张量的张量积，以及更广义地说，如何计算逗点的张量积的派生张量（即 Cotor）群。特别是，我们确定了存在 Cotor 的库奈特公式的条件。我们证明，当且仅当一个适当的系数逗点具有微不足道的协作用时，Cotor 群才有一个简单的库奈特定理。这一结果是我们为计算张量组合积的 Cotor 而构建的谱序列的应用。最后，对于某些在拓扑学上特别自然的非琐碎协元族，我们为第 0 次 Cotor 群（即张量积）的库奈特公式的存在提出了必要条件和充分条件。我们给出了拓扑应用，即谱的粉碎积的亚当斯谱序列的 E2 项和谱的粉碎积的胡勒维茨像的后果。

引用次数: 0

Near automorphisms of the complement or the square of a cycle 循环的补集或平方的近自动形态

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-05-25 DOI: 10.1142/s021949882550286x

Jinxing Zhao

Let <math altimg="eq-00001.gif" display="inline"><mi>G</mi></math> be a graph with vertex set <math altimg="eq-00002.gif" display="inline"><mi>V</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math>, <math altimg="eq-00003.gif" display="inline"><mi>f</mi></math> a permutation of <math altimg="eq-00004.gif" display="inline"><mi>V</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math>. Define <math altimg="eq-00005.gif" display="inline"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo>|</mo><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">−</mo><mi>d</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>|</mo></math> and <math altimg="eq-00006.gif" display="inline"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mo>∑</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math>, where the sum is taken over all unordered pairs <math altimg="eq-00007.gif" display="inline"><mi>x</mi></math>, <math altimg="eq-00008.gif" display="inline"><mi>y</mi></math> of distinct vertices of <math altimg="eq-00009.gif" display="inline"><mi>G</mi></math>. Let <math altimg="eq-00010.gif" display="inline"><mi>π</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math> denote the smallest positive value of <math altimg="eq-00011.gif" display="inline"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math> among all permutations <math altimg="eq-00012.gif" display="inline"><mi>f</mi></math> of <math altimg="eq-00013.gif" display="inline"><mi>V</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math>. A permutation <math altimg="eq-00014.gif" display="inline"><mi>f</mi></math> with <math altimg="eq-00015.gif" display="inline"><msub><mrow><mi>δ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></math> is called a near automorphism of <math altimg="eq-00016.gif" display="inline"><mi>G<

设 G 是一个有顶点集 V(G) 的图，f 是 V(G) 的置换。定义 δf(x,y)=|d(x,y)-d(f(x),f(y))| 和 δf(G)=∑δf(x,y)，其中总和取自 G 中所有无序的不同顶点对 x、y。具有 δf(G)=π(G)的置换 f 称为 G 的近自动形。此外，本文还确定了 π(Cn¯) 和 πCn2 。

引用次数: 0

Semirings generated by idempotents 幂等子生成的半环

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-05-25 DOI: 10.1142/s0219498825502846

David Dolžan

We prove that a semiring multiplicatively generated by its idempotents is commutative and Boolean, if every idempotent in the semiring has an orthogonal complement. We prove that a semiring additively generated by its idempotents is commutative, if every idempotent in the semiring has an orthogonal complement and all the nilpotents in the semirings are central. We also provide examples that the assumptions on the existence of orthogonal complements of idempotents and the centrality of nilpotents cannot be omitted.

我们证明，如果由等幂项乘法生成的配线中的每个等幂项都有一个正交补码，那么这个配线就是交换型布尔配线。我们将证明，如果幂级数中的每个幂级数都有一个正交补码，并且幂级数中的所有零点都是中心点，那么由其幂级数加法生成的谓词就是交换谓词。我们还举例说明了不能省略幂级数的正交补码和零点的中心性这两个假设。

引用次数: 0

Algebraic K0 for unpointed categories 无指向类别的代数 K0

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-05-25 DOI: 10.1142/s0219498825502743

Felix Küng

We construct a natural generalization of the Grothendieck group $K_{0}$ to the case of possibly unpointed categories admitting pushouts by using the concept of heaps recently introduced by Brezinzki. In case of a monoidal category, the defined K0 is shown to be a truss. It is shown that the construction generalizes the classical $K_{0}$ of an abelian category as the group retract along the isomorphism class of the zero object. We finish by applying this construction to construct the integers with addition and multiplication as the decategorification of finite sets and show that in this $K_{0} (\underset{̲}{Top})$ one can identify a CW-complex with the iterated product of its cells.

我们利用布雷津斯基（Brezinzki）最近提出的 "堆"（heaps）概念，构建了格罗内迪克群（Grothendieck group K0）的自然广义，使其适用于可能无指向的、允许推出的范畴。在一元范畴的情况下，定义的 K0 被证明是一个桁。结果表明，随着群沿着零对象的同构类缩回，这种构造概括了经典的无性类 K0。最后，我们将这一构造应用于构建整数的加法和乘法，作为有限集的解归类，并证明在这个 K0(Top̲) 中，我们可以将一个 CW 复数与它的单元的迭积相鉴别。

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引用次数: 0

On upper bounds for asymptotic ideal-grade 关于渐近理想级的上界

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-05-23 DOI: 10.1142/s021949882550272x

Saeed Jahandoust

Let <math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>I</mi></math> and <math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi>J</mi></math> be ideals in a Noetherian ring <math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>R</mi></math> and let <math altimg="eq-00006.gif" display="inline" overflow="scroll"><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math> be nonunits in <math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>R</mi></math>. Then <math altimg="eq-00008.gif" display="inline" overflow="scroll"><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is said to be an asymptotic sequence over <math altimg="eq-00009.gif" display="inline" overflow="scroll"><mi>I</mi></math> if <math altimg="eq-00010.gif" display="inline" overflow="scroll"><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>R</mi><mo>≠</mo><mi>R</mi></math> and if for all <math altimg="eq-00011.gif" display="inline" overflow="scroll"><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math>, <math altimg="eq-00012.gif" display="inline" overflow="scroll"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> is not in any associated prime of the integral closure <math altimg="eq-00013.gif" display="inline" overflow="scroll"><mover accent="false"><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>m</mi></mrow></msup></mrow><mo accent="true">¯</mo></mover></math> of <math altimg="eq-00014.gif" display="inline" overflow="scroll"><msup><mrow><mo stretchy="false">(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><mrow><mi>m</mi></mrow></msup><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>I</mi><mo>,</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi>m</mi></mrow></msup><mi>R</mi></math>, where <math altimg="eq-00015.gi

设 I 和 J 是诺特环 R 中的理想，设 x1,...,xn 是 R 中的非单元。如果 (I,(x1,...,xn))R≠R 并且对于所有 1≤i≤n, xi 不在 (Ii-1)m=(I,(x1,...,xi-1))mR 的积分闭包 (Ii-1)m¯ 的任何相关素数中，其中 m∈ℕ 非常大，那么 x1,...,xn 可以说是 I 上的渐近序列。设 agdI(J) 是 J 中构成 I 上渐近序列的元素的最大数目：(i) ℓ(J)，J 的解析展宽，当 R 是局部时；(ii) ht(I+J)-agd(I)，其中 agd(I) 是 I 中构成 (0)R 上渐近序列的元素的最大数目，并给出了这些特征的若干后果。最后，如果 R 是局部最大理想ᵒ，那么我们将重新证明 agdI(𝔪) 的已知上限。

引用次数: 0

Semi-generalized co-Bassian groups 半广义的共巴斯群

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-05-21 DOI: 10.1142/s0219498825502809

Andrey R. Chekhlov, Peter V. Danchev, Patrick W. Keef

As a common nontrivial generalization of the notion of a generalized co-Bassian group, recently defined by the third author, we introduce the notion of a semi-generalized co-Bassian group and initiate its comprehensive study. Specifically, we give a complete characterization of these groups in the cases of $p$ -torsion groups and groups of finite torsion-free rank by showing that these groups can be completely determined in terms of generalized finite $p$ -ranks and also depends on their quotients modulo the maximal torsion subgroup. Surprisingly, for $p$ -primary groups, the concept of a semi-generalized co-Bassian group is closely related to that of a generalized co-Bassian group.

作为第三位作者最近定义的广义共巴塞尔群概念的一个常见的非难广义化，我们引入了半广义共巴塞尔群的概念，并开始了对它的全面研究。具体地说，我们给出了这些群在 p-扭转群和有限无扭转秩群情况下的完整特征，证明这些群完全可以用广义有限 p-秩来确定，而且还取决于它们的商模数最大扭转子群。令人惊讶的是，对于 p 阶群，半广义共巴斯群的概念与广义共巴斯群的概念密切相关。

引用次数: 0

Morita equivalence and globalization for partial Hopf actions on nonunital algebras 莫里塔等价性与非空格布拉上部分霍普夫作用的全局化

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-05-17 DOI: 10.1142/s0219498825502627

Marcelo Muniz Alves, Tiago Luiz Ferrazza

In this work, we investigate partial actions of a Hopf algebra $H$ on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both $𝕜$ -algebras with local units and $𝕜$ -categories. We show that our partial actions correspond to nonunital algebras in the category of partial representations of $H$ . The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of $𝕜$ -categories.

在这篇论文中，我们研究了霍普夫代数 H 在非空格代数上的部分作用以及相关的部分粉碎乘积，目的是提供一个框架，在这个框架中，我们既可以得到有局部单元的𝕜代数的结果，也可以得到𝕜范畴的结果。我们详细研究了部分作用的全局化存在性这一核心问题，并为一般关联代数的最小全局化的存在性（和唯一性）提供了充分条件。我们扩展了阿巴迪、多库恰耶夫、埃塞尔和西蒙以前的成果，定义了部分霍普夫作用的莫里塔等价性，并证明如果两个对称的部分作用是莫里塔等价的，那么它们的标准全局化也是莫里塔等价的。特别是在具有局部单元的代数上的部分作用的情况下，我们得到了关于代数的部分粉碎乘积的模块类别和𝕜类别的部分粉碎乘积的等价性的几个强有力的结果。

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引用次数: 0

Wilf inequality is preserved under gluing of semigroups Wilf 不等式在半群胶合时得到保留

IF 0.8 3区数学 Q3 MATHEMATICS

Journal of Algebra and Its Applications

Pub Date : 2024-04-26 DOI: 10.1142/s021949882550255x

Srishti Singh, Hema Srinivasan

Wilf Conjecture on numerical semigroups is a question posed by Wilf in 1978 and is an inequality connecting the Frobenius number, embedding dimension and the genus of the semigroup. The conjecture is still open in general. We prove that this Wilf inequality is preserved under gluing of numerical semigroups. If the numerical semigroups minimally generated by $A = {a_{1}, \dots, a_{p}}$ and $B = {b_{1}, \dots, b_{q}}$ satisfy the Wilf inequality, then so does their gluing which is minimally generated by $C = k_{1} A ⊔ k_{2} B$ . We discuss the extended Wilf’s Conjecture in higher dimensions for certain affine semigroups and prove an analogous result.

关于数值半群的 Wilf 猜想是 Wilf 于 1978 年提出的一个问题，是连接半群的弗罗贝尼斯数、嵌入维数和属数的不等式。该猜想在一般情况下仍未解决。我们证明，这个 Wilf 不等式在数字半群的胶合作用下得以保留。如果由 A={a1,...ap} 和 B={b1,...bq} 最小生成的数字半群满足 Wilf 不等式，那么由 C=k1A⊔k2B 最小生成的它们的胶合也满足 Wilf 不等式。我们讨论了某些仿射半群在更高维度上的扩展 Wilf 猜想，并证明了一个类似的结果。

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