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Search: MSC category 16 ( Associative rings and algebras )

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1. CMB Online first

Nasseh, Saeed
On the Generalized Auslander-Reiten Conjecture under Certain Ring Extensions
We show under some conditions that a Gorenstein ring $R$ satisfies the Generalized Auslander-Reiten Conjecture if and only if so does $R[x]$. When $R$ is a local ring we prove the same result for some localizations of $R[x]$.

Keywords:Auslander-Reiten conjecture, finitistic extension degree, Gorenstein rings
Categories:13D07, 16E30, 16E65

2. CMB Online first

Nasseh, Saeed
On the Generalized Auslander-Reiten Conjecture under Certain Ring Extensions
We show under some conditions that a Gorenstein ring $R$ satisfies the Generalized Auslander-Reiten Conjecture if and only if so does $R[x]$. When $R$ is a local ring we prove the same result for some localizations of $R[x]$.

Keywords:Auslander-Reiten conjecture, finitistic extension degree, Gorenstein rings
Categories:13D07, 16E30, 16E65

3. CMB Online first

Hou, Ruchen
On Global Dimensions of Tree Type Finite Dimensional Algebras
A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k-$algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum of possible global dimensions is $n-1$.

Keywords:global dimension, tree type finite dimensional $k-$algebra, quiver
Categories:16D40, 16E10, , 16G20

4. CMB 2014 (vol 57 pp. 609)

Nasr-Isfahani, Alireza
Jacobson Radicals of Skew Polynomial Rings of Derivation Type
We provide necessary and sufficient conditions for a skew polynomial ring of derivation type to be semiprimitive, when the base ring has no nonzero nil ideals. This extends existing results on the Jacobson radical of skew polynomial rings of derivation type.

Keywords:skew polynomial rings, Jacobson radical, derivation
Categories:16S36, 16N20

5. CMB 2014 (vol 57 pp. 511)

Gonçalves, Daniel
Simplicity of Partial Skew Group Rings of Abelian Groups
Let $A$ be a ring with local units, $E$ a set of local units for $A$, $G$ an abelian group and $\alpha$ a partial action of $G$ by ideals of $A$ that contain local units. We show that $A\star_{\alpha} G$ is simple if and only if $A$ is $G$-simple and the center of the corner $e\delta_0 (A\star_{\alpha} G) e \delta_0$ is a field for all $e\in E$. We apply the result to characterize simplicity of partial skew group rings in two cases, namely for partial skew group rings arising from partial actions by clopen subsets of a compact set and partial actions on the set level.

Keywords:partial skew group rings, simple rings, partial actions, abelian groups
Categories:16S35, 37B05

6. CMB 2014 (vol 57 pp. 264)

Dai, Li; Dong, Jingcheng
On Semisimple Hopf Algebras of Dimension $pq^n$
Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $pq^n$ over an algebraically closed field of characteristic $0$. This paper proves that $H$ must possess one of the following structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group algebra of group $G$ of order $p$, and $R$ is a semisimple Yetter--Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$.

Keywords:semisimple Hopf algebra, semisolvability, Radford biproduct, Drinfeld double
Category:16W30

7. CMB 2014 (vol 57 pp. 231)

Bagherian, J.
On the Multiplicities of Characters in Table Algebras
In this paper we show that every module of a table algebra can be considered as a faithful module of some quotient table algebra. Also we prove that every faithful module of a table algebra determines a closed subset which is a cyclic group. As a main result we give some information about multiplicities of characters in table algebras.

Keywords:table algebra, faithful module, multiplicity of character
Categories:20C99, 16G30

8. CMB 2013 (vol 57 pp. 72)

Grari, A.
Un Anneau Commutatif associé à un design symétrique
Dans les articles \cite{1}, \cite{2} et \cite{3}; l'auteur développe une représentation d'un plan projectif fini par un anneau commutatif unitaire dont les propriétés algébriques dépendent de la structure géométrique du plan. Dans l'article \cite{4}; il étend cette représentation aux designs symétriques. Cependant l'auteur de l'article \cite{7} fait remarquer que la multiplication définie dans ce cas ne peut être associative que si le design est un plan projectif. Dans ce papier on mènera une étude de cette représentation dans le cas des designs symétriques. On y montrera comment on peut faire associer un anneau commutatif unitaire à tout design symétrique , on y précisera certaines de ses propriétés, en particulier, celles qui relèvent de son invariance. On caractérisera aussi les géométries projectives finies de dimension supérieure moyennant cette représentation.

Keywords:projective planes, symmetric designs, commutative rings
Categories:05B05, 16S99

9. CMB 2013 (vol 57 pp. 318)

Huang, Zhaoyong
Duality of Preenvelopes and Pure Injective Modules
Let $R$ be an arbitrary ring and $(-)^+=\operatorname{Hom}_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given.

Keywords:(pre)envelopes, (pre)covers, duality, pure injective modules, character modules
Categories:18G25, 16E30

10. CMB 2013 (vol 57 pp. 506)

Galindo, César
On Braided and Ribbon Unitary Fusion Categories
We prove that every braiding over a unitary fusion category is unitary and every unitary braided fusion category admits a unique unitary ribbon structure.

Keywords:fusion categories, braided categories, modular categories
Categories:20F36, 16W30, 18D10

11. CMB 2013 (vol 57 pp. 159)

Oral, Kürşat Hakan; Özkirişci, Neslihan Ayşen; Tekir, Ünsal
Strongly $0$-dimensional Modules
In a multiplication module, prime submodules have the property, if a prime submodule contains a finite intersection of submodules then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodule. A multiplication module in which every prime submodule is strongly prime will be called strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Q-modules and strongly 0-dimensional modules.

Keywords:strongly 0-dimensional rings, Q-module, Von Neumann regular module
Categories:13C99, 16D10

12. CMB 2012 (vol 57 pp. 51)

Fošner, Ajda; Lee, Tsiu-Kwen
Jordan $*$-Derivations of Finite-Dimensional Semiprime Algebras
In the paper, we characterize Jordan $*$-derivations of a $2$-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R=Uoplus V$ such that both $U$ and $V$ are invariant under $delta$. Moreover, $*$ is the identity map of $U$ and $delta,|_U$ is a derivation, and the Jordan $*$-derivation $delta,|_V$ is inner. We also prove the theorem: Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, $operatorname{char},R e 2$, and let $delta$ be a nonzero Jordan $*$-derivation of $R$. If $delta$ is an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and $delta$ is inner.

Keywords:semiprime algebra, involution, (inner) Jordan $*$-derivation, elementary operator
Categories:16W10, 16N60, 16W25

13. CMB 2012 (vol 56 pp. 584)

Liau, Pao-Kuei; Liu, Cheng-Kai
On Automorphisms and Commutativity in Semiprime Rings
Let $R$ be a semiprime ring with center $Z(R)$. For $x,y\in R$, we denote by $[x,y]=xy-yx$ the commutator of $x$ and $y$. If $\sigma$ is a non-identity automorphism of $R$ such that $$ \Big[\big[\dots\big[[\sigma(x^{n_0}),x^{n_1}],x^{n_2}\big],\dots\big],x^{n_k}\Big]=0 $$ for all $x \in R$, where $n_{0},n_{1},n_{2},\dots,n_{k}$ are fixed positive integers, then there exists a map $\mu\colon R\rightarrow Z(R)$ such that $\sigma(x)=x+\mu(x)$ for all $x\in R$. In particular, when $R$ is a prime ring, $R$ is commutative.

Keywords:automorphism, generalized polynomial identity (GPI)
Categories:16N60, 16W20, 16R50

14. CMB 2011 (vol 56 pp. 564)

Herzog, Ivo
Ziegler's Indecomposability Criterion
Ziegler's Indecomposability Criterion is used to prove that a totally transcendental, i.e., $\Sigma$-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module ${_R}U$ to have an indecomposable local dual $U_R^{\sharp}.$

Keywords:pure injective indecomposable module, local dual, generic module, amalgamation
Categories:16G10, 03C60

15. CMB 2011 (vol 56 pp. 344)

Goodaire, Edgar G.; Milies, César Polcino
Involutions and Anticommutativity in Group Rings
Let $g\mapsto g^*$ denote an involution on a group $G$. For any (commutative, associative) ring $R$ (with $1$), $*$ extends linearly to an involution of the group ring $RG$. An element $\alpha\in RG$ is symmetric if $\alpha^*=\alpha$ and skew-symmetric if $\alpha^*=-\alpha$. The skew-symmetric elements are closed under the Lie bracket, $[\alpha,\beta]=\alpha\beta-\beta\alpha$. In this paper, we investigate when this set is also closed under the ring product in $RG$. The symmetric elements are closed under the Jordan product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.

Categories:16W10, 16S34

16. CMB 2011 (vol 55 pp. 271)

Di Vincenzo, M. Onofrio; Nardozza, Vincenzo
On the Existence of the Graded Exponent for Finite Dimensional $\mathbb{Z}_p$-graded Algebras
Let $F$ be an algebraically closed field of characteristic zero, and let $A$ be an associative unitary $F$-algebra graded by a group of prime order. We prove that if $A$ is finite dimensional then the graded exponent of $A$ exists and is an integer.

Keywords:exponent, polynomial identities, graded algebras
Categories:16R50, 16R10, 16W50

17. CMB 2011 (vol 55 pp. 579)

Ndogmo, J. C.
Casimir Operators and Nilpotent Radicals
It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariants
Categories:16W25, 17B45, 16S30

18. CMB 2011 (vol 55 pp. 260)

Delvaux, L.; Van Daele, A.; Wang, Shuanhong
A Note on the Antipode for Algebraic Quantum Groups
Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra. In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode
Categories:16W30, 46L65

19. CMB 2011 (vol 55 pp. 208)

Valenti, Angela; Zaicev, Mikhail
Abelian Gradings on Upper Block Triangular Matrices
Let $G$ be an arbitrary finite abelian group. We describe all possible $G$-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

Keywords:gradings, upper block triangular matrices
Category:16W50

20. CMB 2010 (vol 54 pp. 237)

Creedon, Leo; Gildea, Joe
The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$
Let $RG$ denote the group ring of the group $G$ over the ring $R$. Using an isomorphism between $RG$ and a certain ring of $n \times n$ matrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order $8$ over any finite field of chracteristic $2$ is determined in terms of split extensions of cyclic groups.

Categories:16U60, 16S34, 20C05, 15A33

21. CMB 2010 (vol 53 pp. 587)

Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq
Hulls of Ring Extensions
We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls $\widehat{Q}_{\mathfrak{qB}}(R)$ and $\widehat{Q}_{\mathfrak{qB}}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital $C^*$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of $B$ are strongly Morita equivalent as $C^*$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided.

Keywords:(FI-)extending, Morita equivalent, ring of quotients, essential overring, (quasi-)Baer ring, ring hull, u.p.-monoid, $C^*$-algebra
Categories:16N60, 16D90, 16S99, 16S50, 46L05

22. CMB 2010 (vol 53 pp. 223)

Chuang, Chen-Lian; Lee, Tsiu-Kwen
Density of Polynomial Maps
Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.

Keywords:density, polynomial, endomorphism ring, PI
Categories:16D60, 16S50

23. CMB 2009 (vol 53 pp. 321)

Lee, Tsiu-Kwen; Zhou, Yiqiang
A Theorem on Unit-Regular Rings
Let $R$ be a unit-regular ring and let $\sigma $ be an endomorphism of $R$ such that $\sigma (e)=e$ for all $e^2=e\in R$ and let $n\ge 0$. It is proved that every element of $R[x \mathinner;\sigma]/(x^{n+1})$ is equivalent to an element of the form $e_0+e_1x+\dots +e_nx^n$, where the $e_i$ are orthogonal idempotents of $R$. As an application, it is proved that $R[x \mathinner; \sigma ]/(x^{n+1})$ is left morphic for each $n\ge 0$.

Keywords:morphic rings, unit-regular rings, skew polynomial rings
Categories:16E50, 16U99, 16S70, 16S35

24. CMB 2009 (vol 53 pp. 230)

Doğruöz, S.; Harmanci, A.; Smith, P. F.
Modules with Unique Closure Relative to a Torsion Theory
We consider when a single submodule and also when every submodule of a module M over a general ring R has a unique closure with respect to a hereditary torsion theory on $\operatorname{Mod}$-R.

Keywords:closed submodule, $UC$-module, torsion theory
Category:16S90

25. CMB 2009 (vol 52 pp. 564)

Jin, Hai Lan; Doh, Jaekyung; Park, Jae Keol
Group Actions on Quasi-Baer Rings
A ring $R$ is called {\it quasi-Baer} if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to $C^*$-algebras. Various examples to illustrate and delimit our results are provided.

Keywords:(quasi-) Baer ring, fixed ring, skew group ring, $C^*$-algebra, local multiplier algebra
Categories:16S35, 16W22, 16S90, 16W20, 16U70
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