Simple Modules and Primitive Ideals of Non-Noetherian Generalized Down-Up Algebras

Generalized down-up algebras were first introduced in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]). Their simple weight modules were classified in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]) in the noetherian case, and in Praton (2007 Praton , I. ( 2007 ). Simple weight modules of non-noetherian generalized down-up algebras . Comm. Algebra 35 : 325 – 337 .[Taylor &; Francis Online] , [Google Scholar]) in the non-noetherian case. Here we concentrate on non-noetherian down-up algebras. We show that almost all simple modules are weight modules. We also classify the corresponding primitive ideals.     

Square-zero derivations of matrix algebras over commutative rings

Let R be a commutative ring with identity, M _n (R) the R-algebra consisting of all n by n matrices over R. In this article, for n ≥ 5 we classify linear maps φ from M _n (R) into itself satisfying φ(x)x + xφ(x) = 0 whenever x ² = 0. We call such maps as square-zero derivations.     

A proof that commutative artinian rings are noetherian

Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.     

On the cyclic homology of commutative algebras over arbitrary ground rings

Abstract

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).     

t-Structures and cotilting modules over commutative noetherian rings

For a commutative noetherian ring \(R\) , we establish a bijection between the resolving subcategories consisting of finitely generated \(R\) -modules of finite projective dimension and the compactly generated t-structures in the unbounded derived category \(\mathcal {D}(R)\) that contain \(R[1]\) in their heart. Under this bijection, the t-structures \((\mathcal U,\mathcal V)\) such that the aisle \(\mathcal U\) consists of objects with homology concentrated in degrees \(<n\) correspond to the \(n\) -cotilting classes in \({{\mathrm{Mod}\text {-}R}}\) . As a consequence of these results, we prove that the little finitistic dimension findim \(R\) of \(R\) equals an integer \(n\) if and only if the direct sum \(\bigoplus _{k=0}^n E_k(R)\) of the first \(n+1\) terms in a minimal injective coresolution \(0\rightarrow R\rightarrow E_0(R)\rightarrow E_1(R)\rightarrow \cdots \) of \(R\) is an injective cogenerator of \({{\mathrm{Mod}\text {-}R}}\) .     

On Jordan ideals and matrix rings

For A, a commutative ring, and results by Costa and Keller characterize certain -normalized subgroups of the symplectic group, via structures utilizing Jordan ideals and the notion of radices. The following work creates a Jordan ideal structure theorem for -graded rings, A₀A₁, and a -graded matrix algebra. The major theorem is a generalization of Costa and Keller’s previous work on matrix algebras over commutative rings.     

Additive cyclic codes over finite commutative chain rings

Additive cyclic codes over Galois rings were investigated in Cao et al. (2015). In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the study in Cao et al. (2015), whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.     

Construction of dialgebras through bimodules over algebras

The varieties of dialgebras (also known as Loday-type algebras) over a given type of algebra have been the subject of multiple recent developments. We provide here a construction of such dialgebra varieties via bimodules over an algebra and a surjective equivariant map. Our construction is equivalent to the KP construction (Kolesnikov–Pozhidaev construction) when departing from the set of linearized identities of the algebra variety. The novel construction simplifies the obtention of the dialgebra equations without forcing a complete linearization of the algebra identities. We illustrate the use of the novel construction providing the dialgebras associated to several varieties of algebras, including those over diverse Lie admissible algebras. We provide some novel explorations on the structure of the dialgebras which are easily articulated through our construction.     

Automorphisms of matrix algebras over commutative rings

The automorphisms of an n × n matrix algebra over a commutative ring can fail to be inner. The extent of this failure, however, is under control. For instance, the commutator of any two automorphisms and the nth power of each of them are necessarily inner.     

A Koszul complex over skew polynomial rings

We construct a Koszul complex in the category of left skew polynomial rings associated with a flat endomorphism that provides a finite free resolution of an ideal generated by a Koszul regular sequence.     

Fixed rings of quantum generalized Weyl algebras

Abstract

Generalized Weyl algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite order automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings including global dimension, the Calabi–Yau property, rigidity, and simplicity.     

Determinants of matrices over commutative finite principal ideal rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring R of a fixed determinant a is determined for all $a\in R$ and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.     

可换环上一般线性李代数在几类典型李代数中的扩代数

研究典型李代数的子代数结构,利用矩阵方法决定了含幺可换环上n级一般线性李代数分别在2n级辛代数,2n级正交代数及2n 1级正交代数中的扩代数.