Peter A. Linnell, Gena Puninski, Patrick F. Smith
We prove that every non-finitely generated projective module over the integral group ring of a polycyclic-by-finite group G is free if and only if G is polycyclic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^*$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupo…
On the structure of repeated-root polycyclic codes over local rings
本文针对局部环上重根多循环码的结构问题，提出广义Mattson Solomon多项式实现显式分解完善单根研究，结合矩阵乘积码阐述有限域上该类码的结构性质，并通过两种方法描述其$\bot_0$-对偶码。
We prove that if $A$ is finitely presented algebra with idempotent $e$ such that $A=AeA=A(1-e)A$ then the algebra $eAe$ is finitely presented.
Congruence-simple multiplicatively idempotent semirings and multiplicative divisibility
该论文解决了关于乘法幂等半环的两个猜想，证明了同余单乘法幂等半环有限，以及有限生成交换乘法可除半环为乘法幂等，完善了半环相关分类理论。
Two nonfinitely based additively idempotent semirings of order four
本文证明了两个特定的四元素加法幂等半环无有限基础，并展示了其半环品种的格中存在无限多个不同品种，揭示了有限基础的半环品种的并集不一定有限基础。
We prove that a commutative parasemifield S is additively idempotent provided that it is finitely generated as a semiring. Consequently, every proper commutative semifield T that is finitely generated as a semiring is either additively constant or additively idempotent. As part of the proof, we use the classification of finitely generated lattice-ordered groups to prove that a certain monoid associated to the parasemifield S has a distinguished geometrical property called prismality.
We build on the description of left congruences on an inverse semigroup in terms of the kernel and trace due to Petrich and Rankin. The notion of an inverse kernel for a left congruence is developed. Various properties of both the trace and inverse-kernel are discussed, in particular that both the trace and inverse-kernel maps are onto $\cap$-homomorphisms. The lattice of left congruences is identified as a subset of the direct product of the lattice of congruences on the idempotents and the lattice of full inverse subsemigroups. We use this to describe the lattice of left congruences on the b…
A nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety
本文给出六元加法幂等半环TR_6的无限等式基，证明其生成极限簇，并研究其与SR_6生成的簇的子簇结构。
We prove that every finite idempotent semigroup (band) is finitely related, which means that the clone of its term operations (i.e. operations induced by words) is determined by finitely many relations. This solves an open problem posed by Peter Mayr in 2013.
Let $Λ$ be an artin algebra and $\mathfrak{A}$ a two-sided idempotent ideal of $Λ$, that is, $\mathfrak{A}$ is the trace of a projective $Λ$-module $P$ in $Λ$. We consider the categories of finitely generated modules over the associated rings $Λ/\mathfrak{A}, Λ$ and $Γ=\mathrm{End}_Λ(P)^{op}$ and study the relationship between their homological properties via the Igusa-Todorov functions.
One proves that each almost local-global semihereditary ring has the stacked basis property and is almost Bezout. If M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annhilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a direct sum of 2-generated ideals whose product is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too.
The finite basis problem for matrix semirings over a two-element additively idempotent semiring
研究了二维加法幂等半环上矩阵半环的有限基性质，证明当半环不同于分配格时，矩阵半环具有有限基。
We compute the rank of the group of central units in the integral group ring $\Z G$ of a finite strongly monomial group $G$. The formula obtained is in terms of the strong Shoda pairs of $G$. Next we construct a virtual basis of the group of central units of $\Z G$ for a class of groups $G$ properly contained in the finite strongly monomial groups. Furthermore, for another class of groups $G$ inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of $\Q G$. Finally, we apply these results to describe finitely many gene…
Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for perfect groups is discussed.
It is proven that finite idempotent left non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation on a set $X$ are determined by a left simple semigroup structure on $X$ (in particular, a finite union of isomorphic copies of a group) and some maps $q$ and $φ_x$ on $X$, for $x\in X$. This structure turns out to be a group precisely when the associated structure monoid is cancellative and all the maps $φ_x$ are equal to an automorphism of this group. Equivalently, the structure algebra $K[M(X,r)]$ is right Noetherian, or in characteristic zero it has to be semiprime. The struct…
Every finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each element a of A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero.
For an arbitrary left Artinian ring $R$, explicit descriptions are given of all the left denominator sets $S$ of $R$ and left localizations $S^{-1}R$ of $R$. It is proved that, up to $R$-isomorphism, there are only finitely many left localizations and each of them is an idempotent localization, i.e. $S^{-1}R\simeq S_e^{-1}R$ and ${\rm ass} (S) = {\rm ass} (S_e)$ where $S_e=\{1,e\}$ is a left denominator set of $R$ and $e$ is an idempotent. Moreover, the idempotent $e$ is unique up to a conjugation. It is proved that the number of maximal left denominator sets of $R$ is finite and does not exce…
A semigroup is completely simple if it has no proper ideals and contains a primitive idempotent. We say that a completely simple semigroup $S$ is a homogeneous completely simple semigroup if any isomorphism between finitely generated completely simple subsemigroups of $S$ extends to an automorphism of $S$. Motivated by the study of homogeneous completely regular semigroups, we obtain a complete classification of homogeneous completely simple semigroups, modulo the group case. As a consequence, all finite regular homogeneous semigroups are described, thus extending the work of Cherlin on homoge…
We make a detailed study of idempotent ideals that are traces of countably generated projective right modules. We associate to such ideals an ascending chain of finitely generated left ideals and, dually, a descending chain of cofinitely generated right ideals. The study of the first sequence allows us to characterize trace ideals of projective modules and to show that projective modules can always be lifted modulo the trace ideal of a projective module. As a consequence we give some new classification results of (countably generated) projective modules over particular classes of semilocal rin…
Torsion factors of commutative monoid semirings
本文研究交换幺半群半环因子的加法性质，证明有限秩自由交换幺半群子幺半群对应的半环因子加法几乎可除当且仅当它是挠的。
Big pure projective modules over commutative noetherian rings: comparison with the completion
该研究针对交换诺特环上的纯投射模分类问题，利用有限生成数据刻画无限生成直和项，拓展经典理论至无限生成情形，还研究了相关幺半群结构并得到子幺半群结论与实现结果。
An $R$-module $V$ over a semiring $R$ lacks zero sums (LZS) if $ x +y = 0 \; \Rightarrow \; x = y = 0$. More generally, asubmodule $W$ of $V$ is "summand absorbing", if $ \forall \, x, y \in V: \ x + y \in W \; \Rightarrow \; x \in W, \; y \in W. $ These relate to tropical algebra and modules over idempotent semirings, as well as modules over semirings of sums of squares. In previous work, we have explored the lattice of summand absorbing submodules of a given LZS module, especially those that are finitely generated, in terms of the lattice-theoretic Krull dimension. In this note we describe t…
Given an arbitrary graph E and any field K, a new class of simple left modules over the Leavitt path algebra L of the graph E over K is constructed by using vertices that emit infinitely many edges. The corresponding annihilating primitive ideals are described and is used to show that these new class of simple L-modules are different from(that is non-isomorphic to) any of the previously known simple modules. Using a Boolean subring of idempotents induced by paths in E, bounds for the cardinality of the set of distinct isomorphism classes of simple L-modules are given. We also append other info…
As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine versions of diagram algebras like affine Hecke algebras of type A and affine Temperley-Lieb algebras in a unifying fashion. Affine cellular algebras include Kleshchev's graded quasihereditary algebras, KLR algebras and various other classes of algebras. In this paper we will study ring theoretical properties of affine cellular algebras. We show that any affine cellular algebra $A$ satisfies a polynomial identity. Furthermore, we show that $A$ can be embed…
A direct sum decomposition theory is developed for direct summands (and complements) of modules over a semiring $R$, having the property that $v+w = 0$ implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it always does holds for free modules over the max-plus semiring and related semirings. In such situations, the direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated projective module is a finite direct sum of summands of $R$ (assuming the mild assumption that $1$ is a finite sum of orthogonal primitive idempotents…
Computing the unit group of a commutative finite $\mathbb{Z}$-algebra
本研究针对交换有限$\u2124$-代数的单位群计算问题，通过分层约化结合指数格计算等方法给出计算生成元与结构的算法，已在SageMath中实现并追踪了时间复杂度。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。