We introduce the notion of a projectively simple ring, which is an infinite-dimensional graded k-algebra A such that every 2-sided ideal has finite codimension in A (over the base field k). Under some (relatively mild) additional assumptions on A, we reduce the problem of classifying such rings (in the sense explained in the paper) to the following geometric question, which we believe to be of independent interest.
Let X is a smooth irreducible projective variety. An automorphism f: X -> X is called wild if it X has no proper f-invariant subvarieties. We conjecture that if X admits a wild automorphism then X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if X is a curve or a surface. In the case where X is an abelian variety, we describe all wild automorphisms of X.
In the last two sections we show that if A is projectively simple and admits a balanced dualizing complex, then Proj(A) is Cohen-Macaulay and Gorenstein.
If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring B(X, L, sigma), which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent interest.
The (co)completeness problem for the (projectively) stable module category of an associative ring is studied. (Normal) monomorphisms and (normal) epimorphisms in such a category are characterized. As an application, we give a criterion for the stable category of a left hereditary ring to be abelian. By a structure theorem of Colby-Rutter, this leads to an explicit description of all such rings.
We give sufficient conditions for the descent construction to be the universal central extension of a twisted form of a split simple Lie algebra over a ring. In particular, the universal central extensions of twisted multiloop Lie tori are obtained by the descent construction.
The Gelfand--Kirillov dimension has gained importance since its introduction as an tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand--Kirillov dimension of simple modules over certain simple rings of differential operators.
Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective", respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the l…
We prove that two-sided tilting complexes, and dualizing complexes, over simple Goldie rings (with some technical conditions) are always shifts of invertible bimodules. This allows us to describe the derived Picard groups of such rings, and to deduce these are Gorenstein (and sometimes even Auslander-Gorenstein Cohen-Macaulay) rings.
In this note, we study the notion of purely infinite simple ring in the case of non-unital rings, and we obtain an analog to Zhang's Dichotomy for $σ$-unital purely infinite simple C*-algebras in the purely algebraic context.
Given a group G acting on a ring R, one can construct the skew group ring R*G. Skew group rings have been studied in depth, but necessary and sufficient conditions for the simplicity of a general skew group ring are not known. In this paper, such conditions are given for certain types of skew group rings, with an emphasis on Von Neumann regular skew group rings. Next the results of the first section are used to construct a class of simple skew group rings. In particular, we obtain more efficient proofs of the simplicity of certain rings constructed by H. Kambara and J. Trlifaj.
The Weyl algebra over a field $k$ of characteristic $0$ is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all $\mathbb{Z}$-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study $\mathbb{Z}$-graded simple rings $A$ of any dimension which have a graded quotient ring of the form $K[t, t^{-1}; σ]$ for a field $K$. Under some further hypotheses, we classify all such $A$ in terms of a new construction of simple rings which we introduce …
In this paper we discuss under which conditions cyclic essential extensions of simple modules over a differential operator ring R[z;d] are Artinian. In particular, we study the case when R is either d-simple or d-primitive. Furthermore, we obtain important results when R is an affine algebra of Kull dimension 2. As an application we characterize the differential operator rings C[x,y][z;d] for which cyclic essential extensions of simple modules are Artinian.
We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a non-associative setting. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.
We determine simplicity criteria in characteristics 0 and $p$ for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal element $z$. In the case where this element exists we give simplicity criteria for the rings obtained by inverting $z$ and the rings obtained by factoring out the ideal generated by $z$. The results are illustrated by numerous examples including higher quantized Weyl algebras and generalizations of some low-dimensional symplectic reflection algebras.
Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S^2=0. We classify all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.
Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(σ,δ)$, we may construct its skew power series ring $R[[x;σ,δ]]$. Due to topological obstructions, even if $δ$ is an \emph{inner} $σ$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x; σ, δ]] \cong R[[x'; σ]]$, as might be expected from the theory of skew polynomial rings; similarly when $σ$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;σ,δ]]…
We apply set-theoretic methods to study projective modules and their generalizations over transfinite extensions of simple artinian rings R. When R is hereditary and of cardinality at most $2^ω$, we prove that the Weak Diamond and CH imply that projectivity of an arbitrary module can be tested at the layer epimorphisms of R.
We study injective hulls of simple modules over differential operator rings $R[θ; d]$, providing necessary conditions under which these modules are locally Artinian. As a consequence we characterize Ore extensions of $S=K[x][θ;σ, d]$ for $d$ a $K$-linear derivation and $σ$ a $K$-linear automorphism of $K[x]$ such that injective hulls of simple $S$-modules are locally Artinian.
We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if $R$ is a purely infinite simple ring, then $K_0(R)^+= K_0(R)$, the monoid of isomorphism classes of finitely generated projective $R$-modules is isomorphic to the monoid obtained from $K_0(R)$ by adjoining a new zero element, and $K_1(R)$ is the abelianization of the group of units of $R$. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we com…
We prove that the Zassenhaus conjecture is true for $PSL(2,8)$ and $PSL(2,17)$. This is a continuation of research initiated by W. Kimmerle, M. Hertweck and C. Höfert.
In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring $R = \bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as automorphisms of the commutant of the neutral component subring $R_e$ in $R$ and of the center of $R_e$. We show that if $R$ is a strongly $G$-graded ring where $R_e$ is maximal commutative in $R$, then $R$ is a simple ring if and only if $R_e$ is $G$-simple (i.e. there are no nontrivial $G$-invariant ideals). We also show that if $R_e$ is commutative (not necessarily maximal com…
In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth in [TW1] and [TW2]. We show that if v is a valuation on a field F with associated valuation ring V and v is defectless in a central simple F-algebra A, and C is a subring of A, then the following are equivalent: (a) C is the gauge ring of some minimal v-gauge on A, i.e., a gauge with the minimal number of simple components of C/J(C); (b) C is integra…
It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End(A_p) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of the endomorphism rings of modules over commutative rings is also obtained.
In a recent paper of Alahmadi, Alkan and Lopez-Permouth, a ring R is defined to have no (simple) middle class if the injectivity domain of any (simple) R-module is the smallest or largest possible. Er, Lopez-Permouth and Sokmez use this idea of restricting the class of injectivity domains to classify rings, and give a partial characterization of rings with no middle class. In this work, we continue the study of the property of having no (simple) middle class. We give a structural description of right Artinian right nonsingular rings with no right middle class. We also give a characterization o…
Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz-Pimsner ring contains a non-zero graded ideal, when a relative Cuntz-Pimsner ring is simple, and when every ideal in a relative Cuntz-Pimsner ring is graded, are given. A "Cuntz-Krieger uniqueness theorem" for relative Cuntz-Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz-Pimsner rings are introduced.
In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-)unital $K$-algebra having enough nonzero idempotents (for example, for a vo…
We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero idempotent $f$, then $R$ is simple if and only if it is graded simple and the center of the corner subring $f R_{eGe} f$ is a field. This is a generalization of a result of E. Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity o…
In Picard-Vessiot theory, the Galois theory for linear differential equations, the Picard-Vessiot ring plays an important role, since it is the Picard-Vessiot ring which is a torsor (principal homogeneous space) for the Galois group (scheme). Like fields are simple rings having only (0) and (1) as ideals, the Picard-Vessiot ring is a differentially simple ring, i.e. a differential ring having only (0) and (1) as differential ideals. Having in mind that the classical Galois theory is a theory of extensions of fields, i.e. of simple rings, it is quite natural to ask whether one can also set up a…
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 $Co_3$.
Let D be a division ring finite dimensional over its center F. The goal of this paper is to prove that for any positive integer n there exists a in D^(n); the n-th multiplicative derived subgroup, such that F(a) is a maximal subfield of D. We also show that a single depth-n iterated additive commutator would generate a maximal subfield of D.
Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerle's conjecture on prime graphs.
We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the McLaughlin sporadic group McL. As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.
In this paper a simple right R-module S over a ring R is called hypersimple if its injective hull E(S) is cyclic, and a ring R is called right hypersimple if every simple right R-module is hypersimple. We initiate a study of these new notions, and revisit Osofsky's work on hypercyclic rings, i.e. rings whose cyclic right modules have cyclic injective hulls.
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.
Matlis showed that the injective hull of a simple module over a commutative Noetherian ring is Artinian. Many non-commutative Noetherian rings whose injective hulls of simple modules are locally Artinian have been extensively studied recently. This property had been denoted by property $(\diamond)$. In this paper we investigate, which non-Noetherian semiprimary commutative quasi-local rings $(R, m)$ satisfy property $(\diamond)$. For quasi-local rings $(R,m)$ with $m^3=0$, we prove a characterisation of this property in terms of the dual space of $Soc(R)$. Furthermore, we show that $(R,m)$ sat…
We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group $M_{24}$. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.
Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.
Star operations are an important tool in multiplicative ideal theory. In this paper we apply a special type of star operation, known as $ν$-operation, to define the notion of right Prüfer $ν$-multiplication order. The latter may be viewed as a natural non-commutative version of Prüfer $ν$-multiplication domain. As one of our main results, we establish that an overring of a right Prüfer $ν$-multiplication order is again a right Prüfer $ν$-multiplication order.
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.
A ring is said to satisfy the $2$-nil-sum property if every non central-unit is the sum of two nilpotents. We prove that a ring satisfies the $2$-nil-sum property iff it is either a simple ring with the $2$-nil-sum property or a commutative local ring with nil Jacobson radical, and we provide an example of a simple ring with the $2$-nil-sum property that is not commutative. Moreover, a simple right Goldie ring has the $2$-nil-sum property iff it is a field.
We investigated the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the simple Mathieu group M11. As a consequence, for this group we confirm the conjecture by Kimmerle about prime graphs.
In this paper, we give a complete characterization of Leavitt path algebras which are graded $Σ$-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $Σ$-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical characterization of such a graded $Σ$-$V$ ring $L$% . When the graph $E$ is…
We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M12. As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.
This paper studies similarities and differences between the classes of rings over which each simple module is injective and rings over which each simple module is $Σ$-injective. The rings in the former class are called $V$-rings and the rings in the latter class are called $Σ$-$V$ rings. We have obtained analogues of various well-known results about $V$-rings for $Σ$-$V$ rings. Motivated by a conjecture of Kaplansky, Fisher asked if a prime right $V$-ring is right primitive. Although a counter-example to Kaplansky's conjecture was constructed long ago but Fisher's question is still open. In th…
Let $R$ be a commutative Noetherian ring and $α$ an automorphism of $R$. This paper addresses the question: when does the skew polynomial ring $S = R[θ; α]$ satisfy the property $(\diamond)$, that for every simple $S$-module $V$ the injective hull $E_S(V)$ of $V$ has all its finitely generated submodules Artinian. The question is largely reduced to the special case where $S$ is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when $R$ is an affine algebra over a field…
We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M23 using the Luthar-Passi method. This work is a continuation of the research that we carried out for Mathieu groups M11 and M12. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.
For a unital ring, it is an open question whether flatness of simple modules implies all modules are flat and thus the ring is von Neumann regular. The question was raised by Ramamurthi over 40 years ago who called such rings SF-rings (i.e., simple modules are flat). In this note we show that a SF Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group, has an aperiodic unit space. For graph groupoids this implies that the graphs are acyclic. Combining with the Abrams-Rangaswamy Theorem, it follows that SF Leavitt path algebras are regular, answering Ramamurthi's question …
Every quasi completely regular semiring is a b-lattice of completely Archimedean semirings, i.e., a b-lattice of nil-extensions of completely simple semirings. In this paper we consider the semiring which is a b-lattice of nil-extensions of orthodox completely simple semirings.
For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I of B being A-invariant, that is satisfying AI \subseteq IA. We show that A-simplicity of B is a necessary condition for simplicity of A for a large class of ring extensions when B is a direct summand of A. To obtain sufficient conditions for simplicity of A, we introduce the concept of a degree map for A/B. By this we mean a map d from A to the set of non-n…
We construct finitely generated simple algebras with prescribed growth types, which can be arbitrarily taken from a large variety of (super-polynomial) growth types. This (partially) answers a question raised by the author in a recent paper. Our construction goes through a construction of finitely generated just-infinite, primitive monomial algebras with prescribed growth type, from which we construct uniformly recurrent infinite words with subword complexity having the same growth type. We also discuss the connection between entropy of algebras and their homomorphic images, as well as the deg…
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.
Let $RG$ denote the group ring of the torsion group $G$ over a commutative ring $R$ with identity. In this paper we present proofs of some statements that appear without to be proved in the literature. We establish the valid implications between the ring-theoretic conditions duo, reversible, SI property and symmetric in the setting of group rings. We further show that if the group ring $RG$ possesses any of these properties, then $G$ is a Hamiltonian group and the characteristic of $R$ is either $0$ or $2$. Moreover, we characterize the same properties in group rings $RG$ in the following case…
Two are the objectives of the present paper. First we study properties of a differentially simple commutative ring R with respect to a set D of derivations of R. Among the others we investigate the relation between the D-simplicity of R and that of the local ring RP with respect to a prime ideal P of R and we prove a criterion about the D- simplicity of R in case where R is a 1-dimensional (Krull dimension) finitely generated algebra over a field of characteristic zero and D is a singleton set. The above criterion was quoted without proof in an earlier paper of the author. Second we construct …
A classical result of Claborn states that every abelian group is the class group of a commutative Dedekind domain. Among noncommutative Dedekind prime rings, apart from PI rings, the simple Dedekind domains form a second important class. We show that every abelian group is the class group of a noncommutative simple Dedekind domain. This solves an open problem stated by Levy and Robson in their recent monograph on hereditary Noetherian prime rings.
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.
