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In 1984, Regev started the quantitative study of the space of central polynomials by computing the exponential rate of growth of central polynomials of matrix algebras. More generally, for n ≥ 1 , one considers the dimension c n δ (A) of the space of multilinear central polynomials of degree n modulo the polynomial identities of an algebra A. In 2018, Giambruno and Zaicev proved the limit lim n → ∞ c n δ (A) n exists and it is an integer. In this paper we consider such a situation for superalgebras endowed with a superinvolution or a graded involution and present the existence of the corresponding limit. [ABSTRACT FROM AUTHOR]

Assume that P and Q are elements of A1 satisfying [ P , Q ] = 1 . The Dixmier Conjecture for A1 says that they always generate A1. We show that if P is a sum of not more than 4 homogeneous elements of A1 then P and Q generate A1, which generalizes the main result in [10]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the attached prime ideals of inverse polynomial modules over skew Ore polynomials. [ABSTRACT FROM AUTHOR]

We present a geometrically oriented classification theory for non-Abelian extensions of groupoids generalizing the classification theory for Abelian extensions of groupoids by Westman as well as the familiar classification theory for non-Abelian extensions of groups by Schreier and Eilenberg-MacLane. As an application of our techniques we demonstrate that each extension of groupoids N → E → G gives rise to a groupoid crossed product of G by the groupoid ring of N which recovers the groupoid ring of E up to isomorphism. Furthermore, we make the somewhat surprising observation that our classification methods naturally transfer to the class of groupoid crossed products, thus providing a classification theory for this class of rings. Our study is motivated by the search for natural examples of groupoid crossed products. [ABSTRACT FROM AUTHOR]

We investigate properties of group gradings on matrix rings $ M_n(R) $ M n (R) , where R is an associative unital ring and n is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on $ M_n(R) $ M n (R) is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that $ M_n(R) $ M n (R) is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where R has IBN, we provide a characterization of when $ M_n(R) $ M n (R) is an epsilon-crossed product. Our results are illustrated by several examples. [ABSTRACT FROM AUTHOR]

AbstractUsing the natural Z-grading, Graded Simplicity and Graded Uniqueness Theorems have been obtained in the past for Leavitt path algebras over fields and commutative unital rings. In this paper we extend some of these results to the more general setting of Leavitt path algebras over Clifford semifield and k-semifield. Clifford semifield and k-semifield are natural generalizations of bounded distributive lattices as well as of fields. As evident from some previous works, Leavitt path algebra in these settings is not exactly the same as that in case of fields and commutative rings. Here we characterize graded full k-ideal simplicity of Leavitt path algebra over Clifford semifield and graded k-ideal simplicity over k-semifield by the graph theoretic properties of the underlying directed graph. We apply our simplicity theorems on the Leavitt path algebra of rose graph with one petal to establish that S[x,x−1] is graded full k-ideal simple (respectively, graded k-ideal simple) for any Clifford semifield (respectively, k-semifield) S. Finally we obtain versions of Graded Uniqueness Theorem for Leavitt path algebras with coefficients in Clifford semifield and k-semifield. [ABSTRACT FROM AUTHOR]

For two given hypergraphs, we construct an augmentation hypergraph. We establish an algebra homomorphism from the Leavitt path algebra of the augmentation hypergraph to the tensor product of the corresponding Leavitt path algebras of the two given hypergraphs. And we apply the homomorphism to certain weighted Leavitt path algebras of weighted graphs. We find a sufficient condition on weighted graphs such that one realizes the weighted Leavitt path algebras of the augmentation for weighted graphs as the tensor product of the weighted Leavitt path algebras. [ABSTRACT FROM AUTHOR]

We give a classification of the group gradings on M p (K) , here p is a prime number and K is a field that contains a primitive p-th root of unity if char K ≠ p . The gradings are isomorphic to an elementary grading, to a division grading with support isomorphic to Z p or to a division grading with support isomorphic to Z p × Z p . [ABSTRACT FROM AUTHOR]

Let H be a Hopf algebra that is ℤ -graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of H to be a Zhang twist of H. In particular, we introduce the notion of a twisting pair for H such that the Zhang twist of H by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated with quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction. [ABSTRACT FROM AUTHOR]

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G (H R) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * (H R) a graph obtained from G (H R) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * (R e) denotes a subgraph of G * (H R) , induced by R e *. In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t. Also, let R be semisimple Artinian. We prove that if G * (R e) is connected for every nonzero ring component Re of R, then the connected components of G * (H R) are precisely the graphs G * (R e). [ABSTRACT FROM AUTHOR]

We consider a Mikusiński-type convolution algebra C α , including functions with power-type singularities at the origin as well as all functions continuous on [ 0 , ∞) . Algebraic properties of this space are derived, including its ideal structure, filtered and graded structure, and Jacobson radical. Applications to operators of fractional calculus and the associated integro-differential equations are discussed. [ABSTRACT FROM AUTHOR]

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let G be a group, F an algebraically closed field with char (F) = 0 , A a finite dimensional G-graded associative F -algebra and M a G-graded unitary A -bimodule. We proved that if A = M n (F σ [ H ]) with an elementary-canonical G-grading, where H is a finite abelian subgroup of G and σ ∈ Z 2 (H , F *) , then M being irreducible graded implies that there exists a nonzero homogeneous element w ∈ M satisfying M = B w and B w = w B . Another result we proved generalizes the last one: if G is abelian, A is simple graded and M is finitely generated, then there exist nonzero homogeneous elements w 1 , w 2 , ... , w n ∈ M such that M = A w 1 ⊕ A w 2 ⊕ ⋯ ⊕ A w n , where w i A = A w i ≠ 0 for all i = 1 , 2 , ... , n , and each A w i is irreducible. The elements wi's are associated with the irreducible characters of G. We also describe graded bimodules over graded semisimple algebras. And we finish by presenting a Pierce decomposition of the graded Jacobson radical of any finite dimensional F -algebra with a G-grading. [ABSTRACT FROM AUTHOR]

Let A be an associative superalgebra endowed with a pseudoautomorphism p. In this paper we generalize the Wedderburn-Malcev Theorem in this setting and we prove that the sequence of p-codimensions of A is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and the algebra of 2 × 2 upper-triangular matrices with suitable pseudoautomorphisms. [ABSTRACT FROM AUTHOR]

Let A be a connected cochain DG algebra such that H (A) is a Noetherian graded algebra. We give some criteria for A to be homologically smooth in terms of the singularity category, the cone length of the canonical module k and the global dimension of A . For any cohomologically finite DG A -module M, we show that it is compact when A is homologically smooth. If A is in addition Gorenstein, we get CMreg M = depth A A + Ext. reg M < ∞ , where CMreg M is the Castelnuovo-Mumford regularity of M, depth A A is the depth of A and Ext. reg M is the Ext-regularity of M. [ABSTRACT FROM AUTHOR]

In this paper, we define and study Clifford quadratic complete intersections. After showing some properties of Clifford quantum polynomial algebras, we show that there is a natural one-to-one correspondence between Clifford quadratic complete intersections and commutative quadratic complete intersections. We also provide a calculation method for the point varieties of Clifford quadratic complete intersections. As an application, we give a classification of Clifford quadratic complete intersections in three variables in terms of their characteristic varieties. [ABSTRACT FROM AUTHOR]

Let A be a superalgebra over a field F of characteristic zero. We prove tight relations between graded automorphisms, pseudoautomorphisms, superautomorphisms and K-gradings on A, where K is the Klein group. Moreover, we investigate the consequences of such connections within the theory of polynomial identities. In the second part we focus on the superalgebra U ⁢ T n ⁢ ( F ) {UT_{n}(F)} of n × n {n\times n} upper triangular matrices by completely classifying the graded-pseudo-super automorphism that one can define on it. Finally, we compute the ideals of identities of U ⁢ T n ⁢ ( F ) {UT_{n}(F)} endowed with a graded or a pseudo automorphism, for any n, and the ideals of identities with superautomorphism in the cases n = 2 {n=2} and n = 3 {n=3} . [ABSTRACT FROM AUTHOR]

We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park, and Zelmanov from the group graded case to groupoid, and in some cases category, gradings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a characterization of prime partial skew groupoid rings, and in particular of prime groupoid rings, thereby generalizing a classical result by Connell and partially generalizing recent results by Steinberg. [ABSTRACT FROM AUTHOR]

Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring R ⋊ α G , which carries a natural G-grading. We show that there is a one-to-one correspondence between the G-invariant ideals of R and the graded ideals of the G-graded ring R ⋊ α G . We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of R ⋊ α G . We show that every ideal of R ⋊ α G is graded if and only if α has the residual intersection property. Furthermore, if α is induced by a topological partial action θ, then we prove that minimality of θ is equivalent to G-simplicity of R, topological transitivity of θ is equivalent to G-primeness of R, and topological freeness of θ on every closed invariant subset of the underlying topological space is equivalent to α having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra. [ABSTRACT FROM AUTHOR]

It is well-known that DG-enhancements of the unbounded derived category D qc (X) of quasi-coherent sheaves on a scheme X are all equivalent to each other. Here we present an explicit model which leads to applications in deformation theory. In particular, we shall describe three models for derived endomorphisms of a quasi-coherent sheaf F on a finite-dimensional Noetherian separated scheme (even if F does not admit a locally free resolution). Moreover, these complexes are endowed with DG-Lie algebra structures, which we prove to control infinitesimal deformations of F . [ABSTRACT FROM AUTHOR]

In this paper, we provide a new approach to construct monoidal Hom-Hopf algebras. We investigate monoidal Hom-Hopf algebra structure on a left (n, l)-Hom-crossed coproduct structure with a left n-Hom-smash product structure, obtaining Radford [ n , (n , l) ] -biproduct structure theorem. Then, we study a Hom-coaction admissible mapping system to characterize this Radford [ n , (n , l) ] -biproduct structure. Finally, we study the cosemisimplicity of a special Hom-smash coproduct and prove the related Maschke theorem. [ABSTRACT FROM AUTHOR]

Let G be a group. Let R be a G-graded commutative ring and let M be a graded R-module. A proper graded submodule Q of M is called a graded quasi-primary submodule if whenever r ∈ h ⁢ (R) and m ∈ h ⁢ (M) with r ⁢ m ∈ Q , then either r ∈ Gr ((Q : R M)) or m ∈ Gr M ⁡ (Q) . The graded quasi-primary spectrum qp. Spec g (M) is defined to be the set of all graded quasi-primary submodules of M. In this paper, we introduce and study a topology on qp. Spec g (M) , called the quasi-Zariski topology, and investigate the properties of this topology and some conditions under which ( qp. Spec g (M) , q. τ g) is a Noetherian, spectral space. [ABSTRACT FROM AUTHOR]

This is a general study of twisted Calabi-Yau algebras that are -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi-Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin-Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi-Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi-Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi-Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

A proper graded ideal PP of a commutative graded ring RR is called graded weakly 1-absorbing primary if whenever x,y,zx,y,z are nonunit homogeneous elements of RR with 0≠xyz∈P0\ne xyz\in P, then either xy∈Pxy\in P or zz is in the graded radical of PP. In this article, we explore more results on graded weakly 1-absorbing primary ideals.

In this article, we give a complete characterization of semigroup graded rings which are graded von Neumann regular. We also demonstrate our results by applying them to several classes of examples, including matrix rings and groupoid graded rings. [ABSTRACT FROM AUTHOR]

Let GG be a group and RR be a GG-graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is a generalization of graded 1-absorbing primary ideal. A proper graded ideal PP of RR is said to be a graded weakly 1-absorbing primary ideal of RR if whenever nonunit elements x,y,z∈h(R)x,y,z\in h\left(R) such that 0≠xyz∈P0\ne xyz\in P, then xy∈Pxy\in P or zn∈P{z}^{n}\in P, for some n∈Nn\in {\mathbb{N}}. Several properties of graded weakly 1-absorbing primary ideals are investigated.

Projective modules are a link between geometry and algebra as established by the theorem of Serre-Swan. In this paper, we define the super analog of projective modules and explore this link in the case of some particular super geometric objects. We consider the tangent bundle over the supersphere and show that the module of vector field over a supersphere is a super projective module over the ring of supersmooth functions. Also, we discuss a class of super projective modules that can be constructed from a projection map on modules defined over the ring of supersmooth functions over superspheres. [ABSTRACT FROM AUTHOR]

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß–Leclerc–Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura–Pestun in their study of quiver W -algebras. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we characterize certain systems of first-order nonlinear differential equations whose space of solutions is an s ⁢ l 2 ⁢ (C) -module. We prove that such systems, called Ramanujan systems of Rankin–Cohen type, have a special shape and are precisely the ones whose solution space admits a Rankin–Cohen structure. In the second part of the paper, we consider triangle groups Δ ⁢ (n , m , ∞) . By means of modular embeddings, we associate to every such group a number of systems of nonlinear ODEs whose solutions are algebraically independent twisted modular forms. In particular, all rational weight modular forms on Δ ⁢ (n , m , ∞) are generated by the solutions of one such system (which is of Rankin–Cohen type). As a corollary, we find new relations for the Gauss hypergeometric function evaluated at functions on the upper half-plane. To demonstrate the power of our approach in the non-classical setting, we construct the space of integral weight twisted modular form on Δ ⁢ (2 , 5 , ∞) from solutions of systems of nonlinear ODEs. [ABSTRACT FROM AUTHOR]

In this article, we define a graded trinil clean ring graded by a group. The behavior of the property of being trinil clean is studied in ring constructions of graded group rings, graded matrix rings, and trivial ring extensions. Some sufficient conditions for these rings to be graded trinil clean are given. [ABSTRACT FROM AUTHOR]

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras. [ABSTRACT FROM AUTHOR]

Based on the NilHecke algebra N H n , the odd NilHecke algebra developed by Ellis, Khovanov and Lauda, and on Kang, Kashiwara and Tsuchioka's quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra NH C n as another super-algebraic analogue of N H n . We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text. [ABSTRACT FROM AUTHOR]

Let G be a group, let A = ⊕ g ∈ G A g be an epsilon-strongly graded ring over G, let R := A 1 be the homogeneous component associated with the identity of G, and let 홿횒회횂 ⁢ (R) be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] is an element of 홿횒회횂 ⁢ (R) for all g ∈ G . Moreover, the association g ↦ [ A g ] determines a partial representation of G on 홿횒회횂 ⁢ (R) which induces a partial action γ of G on the center Z ⁢ (R) of R. Sufficient conditions for A to be an Azumaya R γ -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ (A) is the ring of matrices with entries in A, or B = END A (M) = ⊕ l ∈ G Mor A (M , M) l is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A ⁡ (M) where M = A ⊗ R N is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product. [ABSTRACT FROM AUTHOR]

Let U T n (F) be the algebra of the n × n upper triangular matrices and denote U T n (F) (-) the Lie algebra on the vector space of U T n (F) with respect to the usual bracket (commutator), over an infinite field F. In this paper, we give a positive answer to the Specht property for the ideal of the Z n -graded identities of U T n (F) (-) with the canonical grading when the characteristic p of F is 0 or is larger than n - 1 . Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of U T n (F) (-) , is finitely based. Moreover we show that if F is an infinite field of characteristic p = 2 then the Z 3 -graded identities of U T 3 (-) (F) do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of U T 3 (-) (F) , and which is not finitely generated as an ideal of graded identities. [ABSTRACT FROM AUTHOR]

We show the reduced C ∗ -algebra of a graded ample groupoid is a strongly graded C ∗ -algebra if and only if the corresponding Steinberg algebra is a strongly graded ring. We use this to show that the graph C ∗ -algebra of a countable directed graph is a strongly Z -graded C ∗ -algebra if and only if the Leavitt path algebra is a strongly Z -graded ring. The latter of these was shown previously to be equivalent to the graph satisfying property (Y). [ABSTRACT FROM AUTHOR]

Liftable pairs of adjoint functors between braided monoidal categories in the sense of [GV1] provide auto-adjunctions between the associated categories of bialgebras. Motivated by finding interesting examples of such pairs, we study general pre-rigid monoidal categories. Roughly speaking, these are monoidal categories in which for every object X, an object X* and a nicely behaving evaluation map from X* ⊗ X to the unit object exist. A prototypical example is the category of vector spaces over a field, where X* is not a categorical dual if X is not finite-dimensional. We explore the connection with related notions such as right closedness, and present meaningful examples. We also study the categorical frameworks for Turaev's Hopf group-(co)algebras in the light of pre-rigidity and closedness, filling some gaps in literature along the way. Finally, we show that braided pre-rigid monoidal categories indeed provide an appropriate setting for liftability in the sense of loc. cit. and we present an application, varying on the theme of vector spaces, showing how -in favorable cases- the notion of pre-rigidity allows to construct liftable pairs of adjoint functors when right closedness of the category is not available. [ABSTRACT FROM AUTHOR]

Recently, motivated by Anderson, Dumitrescu's S-finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of S-coherent rings, which is the S-version of coherent rings. Let R = ⊕ α ∈ G R α be a commutative ring with unity graded by an arbitrary commutative monoid G, and S a multiplicatively closed subset of nonzero homogeneous elements of R. We define R to be graded-S-coherent ring if every finitely generated homogeneous ideal of R is S-finitely presented. The purpose of this paper is to give the graded version of several results proved in D. Bennis, M. El Hajoui (2018). We show the nontriviality of our generalization by giving an example of a graded-S-coherent ring which is not S-coherent and as a special case of our study, we give the graded version of the Chase's characterization of S-coherent rings. [ABSTRACT FROM AUTHOR]