V. Bovdi, A. Konovalov, S. Linton
Using the Luthar--Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups $Co_1$, $Co_2$ and $Co_3$.
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The Prime Graph Question for integral group rings asks if it is true that if the normalized unit group of the integral group ring of a finite group $G$ contains an element of order $pq$, for some primes $p$ and $q$, also $G$ contains an element of that order. We answer this question for the three Conway sporadic simple groups after reducing it to a combinatorial question about Young tableaus and Littlewood-Richardson coefficients. This finishes work of V. Bovdi, A. Konovalov and S. Linton.
A Ring structure on the Class of Combinatorial Games
本文探讨了组合博弈类在更精细等价关系下的环结构，提出了一种新的等价关系，并指出其他可能的等价关系。
We investigate the possible character values of torsion units of the normalized unit group of the integral group ring of Mathieu sporadic group $M_{22}$. We confirm the Kimmerle conjecture on prime graphs for this group and specify the partial augmentations for possible counterexamples to the stronger Zassenhaus conjecture.
Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Higman-Sims simple sporadic group HS. As a consequence, we confirm the Kimmerle's conjecture on prime graphs for this sporadic group.
Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups $J_1$, $J_2$ and $J_3$ is the same as that of the normalized unit group of their respective integral group ring.
The connections between the properties of associative rings that are Lie-solvable (Engel, n-Engel, locally finite, respectively) and the properties of their adjoin subgroups are investigated.
Let n be an arbitrary natural number. The class of (strongly) n-torsion clean rings is introduced and investigated. Abelian n-torsion clean rings are somewhat characterized and a complete characterization of strongly n-torsion clean rings is given in the case when n is odd. Some open questions are posed at the end.
It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential endomorphism rings are faithful. The paper will appear in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry.
We completely determine those natural numbers $n$ for which the full matrix ring $M_n(F_2)$ and the triangular matrix ring $T_n(F_2)$ over the two elements field $F_2$ are either n-torsion clean or are almost n-torsion clean, respectively. These results somewhat address and settle a question, recently posed by Danchev-Matczuk in Contemp. Math. (2019) as well as they supply in a more precise aspect the nil-cleanness property of the full matrix $n\times n$ ring $M_n(F_2)$ for all naturals $n \geq 1$, established in Linear Algebra and Appl. (2013) by Breaz-Calugareanu-Danchev-Micu and again in Li…
We show that for every prime $r$ all $r$-subgroups in the normalized units of the integral group ring of $\operatorname{PSL}(2,p^3)$ are isomorphic to subgroups of $\operatorname{PSL}(2,p^3)$. This answers a question of M. Hertweck, C.R. Höfert and W. Kimmerle for this series of groups.
For $c\geq 2$, the free centre-by-(nilpotent-of-class-c-1)-by abelian Lie ring on a set $X$ is the quotient $L/[(L')^c,L]$ where $L$ is the free Lie ring on $X$, and $(L')^c$ denotes the $c$th term of the lower central series of the derived ideal $L'=L^2$ of $L$. In this paper we give a complete description of the torsion subgroup of its additive group in the case where $|X|=2$ and $c$ is a prime number.
It is shown that for any torsion unit of augmentation one in the integral group ring $\mathbb{Z} G$ of a finite solvable group $G$, there is an element of $G$ of the same order.
We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p-block of a finite group G the normalized unit group of the integral group ring of G contains an element of order pq if and only if so does G, for q a prime different from p. Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.
Units, zero-divisors and idempotents in rings graded by torsion-free groups
本文将无挠群群环的三大经典Kaplansky问题推广至无挠群分次环，限定研究无非平凡齐次零因子的分次环，解决了唯一积群分次环的情形，证明了扩展问题与经典问题层级一致，并得到分次环的不可分解性等性质，推广了经典群环猜想。
A torsion theoretical characterization of left Noetherian rings $R$ over which injective hulls of simple left modules are locally Artinian is given. Sufficient conditions for a left Noetherian ring to satisfy this finiteness condition are obtained in terms of torsion theories.
Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum-Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is…
Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class ${\mathcal C}$. First, we show that a finitely generated module over a ring from the class ${\mathcal C}$ splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from ${\mathcal C}$ and prove that this dimension has all the nice properties of the di…
For a finite group $G$, let $\tilde{\mathbb{Z}}$ be the semilocalization of $\mathbb{Z}$ at the prime divisors of $|G|$. If $G$ is a Frobenius group with Frobenius kernel $K$, it is shown that each torsion unit in the group ring $\tilde{\mathbb{Z}} G$ which maps to the identity under the natural ring homomorphism $\tilde{\mathbb{Z}} G \rightarrow \tilde{\mathbb{Z}} G/K$ is conjugate to an element of $G$ by a unit in $\tilde{\mathbb{Z}} G$.
Let $Γ$ be a discrete group. Following Linnell and Schick one can define a continuous ring $c(Γ)$ associated with $Γ$. They proved that if the Atiyah Conjecture holds for a torsion-free group $Γ$, then $c(Γ)$ is a skew field. Also, if $Γ$ has torsion and the Strong Atiyah Conjecture holds for $Γ$, then $c(Γ)$ is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group $Γ=Z_2\wr Z$. It is known that $C(Z_2\wr Z)$ does not even have a classical ring of quotients. Our main result is that if $H$ is amenable, then $c(Z_2\wr H)$ is isomor…
We study the notions of $n$-hereditary rings and its connection to the classes of finitely $n$-presented modules, FP$_n$-injective modules, FP$_n$-flat modules and $n$-coherent rings. We give characterizations of $n$-hereditary rings in terms of quotients of injective modules and submodules of flat modules, and a characterization of $n$-coherent using an injective cogenerator of the category of modules. We show two torsion pairs with respect to the FP$_n$-injective modules and the FP$_n$-flat modules over $n$-hereditary rings. We also provide an example of a Bézout ring which is 2-hereditary, …
A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsion theories are higher differential as well. Then, we study conditions under which a higher derivation extended to a right module of quotients extends also to a right module of quotients with respect to a…
We generalize the tilting process by Happel, Reiten and Smalø to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones to all right noetherian rings.
Endomorphism algebras over commutative rings and torsion in self tensor products
本文研究交换 Noetherian 局部环上有限生成模的自同态环作用下的张量积，证明在温和条件下不可分解模的自张量积必有挠元。
We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations t…
This is the second of two papers on the injective spectrum of a right noetherian ring. In the prequel, we considered the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples. In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure s…
In this work we introduce a new concept, namely, $τ_{s}$-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show that for any ring $R$ with unit, if $R$ is purely $τ_{s}$-extending then every cyclic $τ$-nonsingular $R$-module is flat and we show that this fact is true over a principal ideal domain as well. Also, we make a classification for the direct sums of the rings to be purely $τ_{s}$-extending.
Kaplanski's Zero Divisor Conjecture envisions that for a torsion-free group G and an integral domain R, the group ring R[G] does not contain non-trivial zero divisors. We define the length of an element a in R[G] as the minimal non-negative integer k for which there are ring elements r_1,...,r_k in R and group elements g_1,...,g_k in G such that a = r_1 g_1+...+r_k g_k. We investigate the conjecture when R is the field of rational numbers. By a reduction to the finite field with two elements, we show that if ab = 0 for non-trivial elements in the group ring of a torsion-free group over the rat…
We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.
We study Abelian groups $A$ with centrally essential endomorphism ring $\text{End}\,A$. If $A$ is a such group which is either a torsion group or a non-reduced group, then the ring $\text{End}\,A$ is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings. The paper will appear in Communications in Algebra.
H. J. Zassenhaus conjectured that any unit of finite order and augmentation one in the integral group ring of a finite group $G$ is conjugate in the rational group algebra to an element of $G$. One way to verify this is showing that such unit has the same distribution of partial augmentations as an element of $G$ and the HeLP Method provides a tool to do that in some cases. In this paper we use the HeLP Method to describe the partial augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.
In the class of reduced Abelian torsion-free groups $G$ of finite rank, we describe TI-groups, this means that every associative ring on $G$ is filial. If every associative multiplication on $G$ is the zero multiplication, then $G$ is called a $nil_a$-group. It is proved that a reduced Abelian torsion-free group $G$ of finite rank is a $TI$-group if and only if $G$ is a homogeneous Murley group or $G$ is a $nil_a$-group. We also study the interrelations between the class of homogeneous Murley groups and the class of $nil_a$-groups. For any type $t\ne (\infty,\infty,\ldots)$ and every integer $…
A proof of a theorem of M. Hertweck presented during a seminar in January 2013 in Stuttgart is given. The proof is based on a preprint given to me by Hertweck. Let $R$ be a commutative ring, $G$ a finite group, $N$ a normal $p$-subgroup of $G$ and denote by $RG$ the group ring of $G$ over $R$. It is shown that a torsion unit $u$ in $\mathbb{Z}G$ mapping to the identity under the natural homomorphism $\mathbb{Z}G \rightarrow \mathbb{Z}G/N$ is conjugate in the unit group of $\mathbb{Z}_pG$ to an element in $N$. Here $\mathbb{Z}_p$ denotes the $p$-adic integers. The result is achieved proving a r…
For a torsion unit $u$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$, and a prime $p$ which does not divide the order of $u$ (but the order of $G$), a relation between the partial augmentations of $u$ on the $p$-regular classes of $G$ and Brauer character values is noted, analogous to the obvious relation between partial augmentations and ordinary character values. For non-solvable $G$, consequences concerning rational conjugacy of $u$ to a group element are discussed, considering as examples the symmetric group $S_{5}$ and the groups $\text{\rm PSL}(2,p^{f})$.
We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group $\operatorname{PSL}(2,19)$. We also prove the Zassenhaus Conjecture for $\operatorname{PSL}(2,23)$. In a second application we show that there are no normalized units of order $6$ in the integral group rings of $M_{10}$ and $\operatorname{PGL}(2,9)$. This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative an…
We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation of the center of the resulting deformed ring. This connects with earlier calculations by Vafa and Witten of chiral numbers and deformations of a similar orbifold.
In this paper, we describe all automorphism-liftable torsion modules over non-primitive hereditary Noetherian prime rings. We also study automorphism-liftable non-torsion modules over not necessarily commutative Dedekind prime rings
Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. Let $Z_2(M)$ be the second singular submodule of $M$. In this paper, we define Goldie Rickart modules by utilizing the endomorphisms of a module. The module $M$ is called Goldie Rickart if for any $f\in S$, $f^{-1}(Z_2(M))$ is a direct summand of $M$. We provide several characterizations of Goldie Rickart modules and study their properties. Also we present that semisimple rings and right $Σ$-$t$-extending rings admit some characterizations in terms of Goldie Rickart modules.
We consider some existing results regarding rings for which the classes of torsion-free and non-singular right modules coincide. Here, a right $R$-module $M$ is non-singular if $xI$ is nonzero for every nonzero $x \in M$ and every essential right ideal $I$ of $R$, and a right $R$-module $M$ is torsion-free if $Tor_{1}^{R}(M,R / Rr)=0$ for every $r \in R$. In particular, we consider a ring $R$ for which the classes of torsion-free and non-singular right $S$-modules coincide for every ring $S$ Morita-equivalent to $R$. We make use of these results, as well as the existence of a Morita-equivalenc…
Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring with total quotient ring $K$. An $R$-module $M$ is called simple divisible, if $M$ is divisible $\neq 0$, but every proper submodule $0 \neq U \subsetneqq M$ is not divisible. Dually, $M$ is called simple torsion free, if $M$ ist torsion free $\neq 0$, but, for every proper submodule $0 \neq U \subsetneqq M$, the factor module $M/U$ is not torsion free. Our first result is that $M \neq 0$ is simple torsion free iff $M$ is a submodule of $κ(\mathfrak{p}) = R_{\mathfrak{p}}/\mathfrak{p} R_{\mathfrak{p}}$ for a maximal element $\mathf…
Let $k$ be a field of characteristic different from $2$ and let $G$ be a nonabelian residually torsion-free nilpotent group. It is known that $G$ is an orderable group. Let $k(G)$ denote the subdivision ring of the Malcev-Neumann series ring generated by the group algebra of $G$ over $k$. If $\ast$ is an involution on $G$, then it extends to a unique $k$-involution on $k(G)$. We show that $k(G)$ contains pairs of symmetric elements with respect to $\ast$ which generate a free group inside the multiplicative group of $k(G)$. Free unitary pairs also exist if $G$ is torsion-free nilpotent. Finall…
We investigate conditions for when the $t$-structure of Happel-Reiten-Smalø associated to a torsion pair is a compactly generated $t$-structure. The concept of a $t$CG torsion pair is introduced and for any ring $R$, we prove that $\mathbf{t}=(\mathcal{T},\mathcal{F})$ is a $t$CG torsion pair in $R\text{-Mod}$ if, and only if, there exists, $\{T_λ\}$ a set of finitely presented $R$-modules in $\mathcal{T}$, such that $\mathcal{F}=\bigcap \text{Ker}({\text{Hom}}_{R}(T_λ,?))$. We also show that every $t$CG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give…
Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples
本文研究了环R中平衡对与余 torsion 三元组的关系，证明了右几乎格罗蒙德环的条件，并通过无限维翻转理论分类了所有余 torsion 三元组。
It is known that every nonzero Jordan ideal of $2$-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a $2$-torsion free prime ring contains a nonzero ideal. This can be interpreted by saying that studying identities over one sided ideals is the "optimal" case to study identities. With this fact in mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and action of generalized derivations in rings, Miskolc Mathematical Notes, \textbf{16} (2015), 769 -- 779] to the context of nonzero left ideals.
Let A be a commutative ring, and let \a = \frak{a} be a finitely generated ideal in it. It is known that a necessary and sufficient condition for the derived \a-torsion and \a-adic completion functors to be nicely behaved is the weak proregularity of \a. In particular, the MGM Equivalence holds. Because weak proregularity is defined in terms of elements of the ring (specifically, it involves limits of Koszul complexes), it is not suitable for noncommutative ring theory. In this paper we introduce a new condition on a torsion class T in a module category: weak stability. Our first main theorem …
We study the classes of modules which are generated by a silting module. In the case of either hereditary or perfect rings it is proved that these are exactly the torsion $\mathcal{T}$ such that the regular module has a special $\mathcal{T}$-preenvelope. In particular every torsion enveloping class in $\textrm{Mod-} R$ are of the form $\mathrm{Gen}(T)$ for a minimal silting module $T$. For the dual case we obtain for general rings that the covering torsion-free classes of modules are exactly the classes of the form $\mathrm{Cogen}(T)$, where $T$ is a cosilting module.
Generalized $t$-Fine and Quasi $t$-Fine Rings
本文引入广义$t$-精细环与广义拟$t$-精细环，用挠单位元与（拟）幂零元分解Jacobson根外元素，刻画其性质并研究矩阵环、群环等上的表现。
In the present notes we generalize the classical work of Demazure [Invariants symétriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let G be a split semisemiple linear algebraic group over a field and let T be its split maximal torus. We construct a generalized characteristic map relating the so called formal group ring of the character group of T with the cohomology of the variety of Borel subgroups of G. The main result of the paper says that the kernel of this map is generated by W-invariant elements, where W is the Weyl group o…
We prove that every perfect torsion theory for a ring $R$ is differential (in the sense of [P. E. Bland, Differential torsion theory, Journal of Pure and Applied Algebra 204 (2006) 1 -- 8]). In this case, we construct the extension of a derivation of a right $R$-module $M$ to a derivation of the module of quotients of $M$. Then, we prove that the Lambek and Goldie torsion theories for any $R$ are differential.
Let R be a ring and let T be a hereditary torsion class of R-modules. The inclusion of the localizing subcategory generated by T into the derived category of R has a right adjoint, which is a colocalization. Benson has recently shown how to compute this right sdjoint when R is the group ring of a finite group over a prime field and T is the hereditary torsion class generated by a simple module. We generalize Benson's construction to the case where T is any hereditary torsion class on R. This yields an explicit formula for the colocalization with respect to T, using an injective cogenerator for…
Let $\mathbb Z \langle X \rangle$ be the free unital associative ring freely generated by an infinite countable set $X = \{ x_1,x_2, \dots \}$. Define a left-normed commutator $[x_1,x_2, \dots, x_n]$ by $[a,b] = ab - ba$, $[a,b,c] = [[a,b],c]$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $\mathbb Z \langle X \rangle$ generated by all commutators $[a_1,a_2, \dots, a_n]$ $( a_i \in \mathbb Z \langle X \rangle )$. Let $T^{(3,2)}$ be the two-sided ideal of the ring $\mathbb Z \langle X \rangle$ generated by all elements $[a_1, a_2, a_3, a_4]$ and $[a_1, a_2] [a_3, a_4, a_5]$ $(a_i \in \…
Torsion modules and differential operators in infinitely many variables
本文探讨了无限生成理想下的torsion模概念，提出微分算子的超限阶和局部阶定义，并证明任意序数均可作为无限多个变量多项式代数上微分算子的阶数。
Let $K$ be a 2-torsion free ring with identity. We give a description of the Lie derivations of $R=R_{n}(K,J)=NT_{n}(K)+M_{n}(J)$, the ring of all $n\times n$ matrices over $K$ such that the entries on and above the main diagonal are elements of an ideal $J$ of $K$.
In 1960 Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In this paper we address Fuchs' question for {\it finitely generated abelian} groups and we consider the problem of characterizing those groups which arise in some fixed classes of rings $\mathcal C$, namely the integral domains, the torsion free rings and the reduced rings. To determine the realizable groups we have to establish what finite abelian groups $T$ (up to isomorphism) occur…
Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or of a finite dimensional algebra of finite representation type, can be parametrised by silting modules. In these notes, we give a brief introduction to the fairly recent concepts of silting and cosilting module, and we explain the classification results mentioned above.
Torsion classes of extended Dynkin quivers over commutative rings
本文研究了交换环上代数的 torsion classes 相容性，证明了当 $Q$ 为扩展 Dynkin quiver 且 $R$ 为 Dedekind domain 或二维 Noetherian semilocal normal ring 时，路代数 $RQ$ 满足相容性条件。
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