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26. 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, amalgamationCategories:16G10, 03C60

27. 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

28. 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 algebrasCategories:16R50, 16R10, 16W50

29. 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 invariantsCategories:16W25, 17B45, 16S30

30. 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 antipodeCategories:16W30, 46L65

31. 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 matricesCategory:16W50

32. 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

33. 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, $C^*$-algebraCategories:16N60, 16D90, 16S99, 16S50, 46L05

34. 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, PICategories:16D60, 16S50

35. 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 ringsCategories:16E50, 16U99, 16S70, 16S35

36. 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 theoryCategory:16S90

37. 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 algebraCategories:16S35, 16W22, 16S90, 16W20, 16U70

38. CMB 2009 (vol 52 pp. 267)

Ko\c{s}an, Muhammet Tamer
 Extensions of Rings Having McCoy Condition Let $R$ be an associative ring with unity. Then $R$ is said to be a {\it right McCoy ring} when the equation $f(x)g(x)=0$ (over $R[x]$), where $0\neq f(x),g(x) \in R[x]$, implies that there exists a nonzero element $c\in R$ such that $f(x)c=0$. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if $R$ is a right McCoy ring, then $R[x]/(x^n)$ is a right McCoy ring for any positive integer $n\geq 2$ . Keywords:right McCoy ring, Armendariz ring, reduced ring, reversible ring, semicommutative ringCategories:16D10, 16D80, 16R50

39. CMB 2009 (vol 52 pp. 145)

Wang, Z.; Chen, J. L.
 $2$-Clean Rings A ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean rings and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show that for any ring $R$, the endomorphism ring of a free $R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$ of all $\omega\times \omega$ row and column-finite matrices over any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is considered where $R$ is a local ring. Keywords:$2$-clean rings, $2$-good rings, free modules, row and column-finite matrix rings, group ringsCategories:16D70, 16D40, 16S50

40. CMB 2009 (vol 52 pp. 39)

Cimpri\v{c}, Jakob
 A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution. Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometryCategories:16W80, 46L05, 46L89, 14P99

41. CMB 2008 (vol 51 pp. 460)

Smoktunowicz, Agata
 On Primitive Ideals in Graded Rings Let $R=\bigoplus_{i=1}^{\infty}R_{i}$ be a graded nil ring. It is shown that primitive ideals in $R$ are homogeneous. Let $A=\bigoplus_{i=1}^{\infty}A_{i}$ be a graded non-PI just-infinite dimensional algebra and let $I$ be a prime ideal in $A$. It is shown that either $I=\{0\}$ or $I=A$. Moreover, $A$ is either primitive or Jacobson radical. Categories:16D60, 16W50

42. CMB 2008 (vol 51 pp. 424)

Novelli, Jean-Christophe; Thibon, Jean-Yves
 Noncommutative Symmetric Bessel Functions The consideration of tensor products of $0$-Hecke algebra modules leads to natural analogs of the Bessel $J$-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions. Categories:05E05, 16W30, 05A15

43. CMB 2008 (vol 51 pp. 291)

Spinelli, Ernesto
 Group Algebras with Minimal Strong Lie Derived Length Let $KG$ be a non-commutative strongly Lie solvable group algebra of a group $G$ over a field $K$ of positive characteristic $p$. In this note we state necessary and sufficient conditions so that the strong Lie derived length of $KG$ assumes its minimal value, namely $\lceil \log_{2}(p+1)\rceil$. Keywords:group algebras, strong Lie derived lengthCategories:16S34, 17B30

44. CMB 2008 (vol 51 pp. 261)

Neeb, Karl-Hermann
 On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups An $n$-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field. Keywords:quantum torus, normal form, automorphisms of quantum toriCategory:16S35

45. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
 Homotopy Formulas for Cyclic Groups Acting on Rings The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff. Keywords:group cohomology, norm map, cyclic group, homotopyCategories:20J06, 20K01, 16W22, 18G35

46. CMB 2007 (vol 50 pp. 105)

Klep, Igor
 On Valuations, Places and Graded Rings Associated to $*$-Orderings We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups. Keywords:$*$--orderings, valuations, rings with involutionCategories:14P10, 16S30, 16W10

47. CMB 2006 (vol 49 pp. 347)

Ecker, Jürgen
 Affine Completeness of Generalised Dihedral Groups In this paper we study affine completeness of generalised dihedral groups. We give a formula for the number of unary compatible functions on these groups, and we characterise for every $k \in~\N$ the $k$-affine complete generalised dihedral groups. We find that the direct product of a $1$-affine complete group with itself need not be $1$-affine complete. Finally, we give an example of a nonabelian solvable affine complete group. For nilpotent groups we find a strong necessary condition for $2$-affine completeness. Categories:08A40, 16Y30, 20F05

48. CMB 2006 (vol 49 pp. 265)

Nicholson, W. K.; Zhou, Y.
 Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in C[x]$, Camillo and Sim\'{o}n called a ring$g(x)$-clean if every element is the sum of a unit and a root of$g(x)$. If$V$is a vector space of countable dimension over a division ring$D,$they showed that$\end {}_{D}V$is$g(x)$-clean provided that$g(x)$has two roots in$C(D)$. If$g(x)=x-x^{2}$this shows that$\end {}_{D}V$is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that$\Mend {}_{R}M$is$g(x)$-clean for any semisimple module$M$over an arbitrary ring$R$provided that$g(x)\in (x-a)(x-b)C[x]$where$a,b\in C$and both$b$and$b-a$are units in$R$. Keywords:Clean rings, linear transformations, endomorphism ringsCategories:16S50, 16E50 49. CMB 2005 (vol 48 pp. 587) Lopes, Samuel A.  Separation of Variables for$U_{q}(\mathfrak{sl}_{n+1})^{+}$Let$U_{q}(\SL)^{+}$be the positive part of the quantized enveloping algebra$U_{q}(\SL)$. Using results of Alev--Dumas and Caldero related to the center of$U_{q}(\SL)^{+}$, we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra$U(\g)$of a complex semisimple Lie algebra$\g$, and also of an analogous result of Joseph--Letzter for the quantum algebra$\Check{U}_{q}(\g)$. Of greater importance to its representation theory is the fact that$\U{+}$is free over a larger polynomial subalgebra$N$in$n$variables. Induction from$N$to$\U{+}\$ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules. Categories:17B37, 16W35, 17B10, 16D60

50. CMB 2005 (vol 48 pp. 355)

Chebotar, M. A.; Ke, W.-F.; Lee, P.-H.; Shiao, L.-S.
 On Maps Preserving Products Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view. Categories:16W20, 16N50, 16N60
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