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51. CMB 2005 (vol 48 pp. 80)

Herman, Allen; Li, Yuanlin; Parmenter, M. M.
Trivial Units for Group Rings with $G$-adapted Coefficient Rings
For each finite group $G$ for which the integral group ring $\mathbb{Z}G$ has only trivial units, we give ring-theoretic conditions for a commutative ring $R$ under which the group ring $RG$ has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if $R$ is a ring of finite characteristic and $RG$ has only trivial units, then $G$ has order at most 3.

Categories:16S34, 16U60, 20C05

52. CMB 2004 (vol 47 pp. 343)

Drensky, Vesselin; Hammoudi, Lakhdar
Combinatorics of Words and Semigroup Algebras Which Are Sums of Locally Nilpotent Subalgebras
We construct new examples of non-nil algebras with any number of generators, which are direct sums of two locally nilpotent subalgebras. Like all previously known examples, our examples are contracted semigroup algebras and the underlying semigroups are unions of locally nilpotent subsemigroups. In our constructions we make more transparent than in the past the close relationship between the considered problem and combinatorics of words.

Keywords:locally nilpotent rings,, nil rings, locally nilpotent semigroups,, semigroup algebras, monomial algebras, infinite words
Categories:16N40, 16S15, 20M05, 20M25, 68R15

53. CMB 2004 (vol 47 pp. 445)

Pirkovskii, A. Yu.
Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators
For a locally compact group $G$, the convolution product on the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang \cite{Neuf_PhD}. We study homological properties of the convolution algebra $\nN(L^p(G))$ and relate them to some properties of the group $G$, such as compactness, finiteness, discreteness, and amenability.

Categories:46M10, 46H25, 43A20, 16E65

54. CMB 2003 (vol 46 pp. 14)

Bahturin, Yu. A.; Parmenter, M. M.
Generalized Commutativity in Group Algebras
We study group algebras $FG$ which can be graded by a finite abelian group $\Gamma$ such that $FG$ is $\beta$-commutative for a skew-symmetric bicharacter $\beta$ on $\Gamma$ with values in $F^*$.

Categories:16S34, 16R50, 16U80, 16W10, 16W55

55. CMB 2002 (vol 45 pp. 499)

Bahturin, Yu. A.; Zaicev, M. V.
Group Gradings on Matrix Algebras
Let $\Phi$ be an algebraically closed field of characteristic zero, $G$ a finite, not necessarily abelian, group. Given a $G$-grading on the full matrix algebra $A = M_n(\Phi)$, we decompose $A$ as the tensor product of graded subalgebras $A = B\otimes C$, $B\cong M_p (\Phi)$ being a graded division algebra, while the grading of $C\cong M_q (\Phi)$ is determined by that of the vector space $\Phi^n$. Now the grading of $A$ is recovered from those of $A$ and $B$ using a canonical ``induction'' procedure.

Category:16W50

56. CMB 2002 (vol 45 pp. 711)

Yoshii, Yoji
Classification of Quantum Tori with Involution
Quantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type $\rmA_1$, $\rmC$ and $\BC$. We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type $\rmC$.

Category:16W50

57. CMB 2002 (vol 45 pp. 451)

Allison, Bruce; Smirnov, Oleg
Coordinatization Theorems For Graded Algebras
In this paper we study simple associative algebras with finite $\mathbb{Z}$-gradings. This is done using a simple algebra $F_g$ that has been constructed in Morita theory from a bilinear form $g\colon U\times V\to A$ over a simple algebra $A$. We show that finite $\mathbb{Z}$-gradings on $F_g$ are in one to one correspondence with certain decompositions of the pair $(U,V)$. We also show that any simple algebra $R$ with finite $\mathbb{Z}$-grading is graded isomorphic to $F_g$ for some bilinear from $g\colon U\times V \to A$, where the grading on $F_g$ is determined by a decomposition of $(U,V)$ and the coordinate algebra $A$ is chosen as a simple ideal of the zero component $R_0$ of $R$. In order to prove these results we first prove similar results for simple algebras with Peirce gradings.

Category:16W50

58. CMB 2002 (vol 45 pp. 388)

Gille, Philippe
Algèbres simples centrales de degré 5 et $E_8$
As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree~5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type~$E_8$.

Keywords:algèbres simples centrales, cohomologie galoisienne
Categories:16S35, 12G05, 20G15

59. CMB 2002 (vol 45 pp. 448)

Zhou, Yiqiang
Erratum: A Characterization of Left Perfect Rings
An error in {\it A characterization of left perfect rings}, Canad. Math. Bull. (3) {\bf 38}(1995), 382--384, is indicated and the consequences identified.

Category:16L30

60. CMB 2002 (vol 45 pp. 11)

Bahturin, Yuri; Kochetov, Mikhail; Montgomery, Susan
Polycharacters of Cocommutative Hopf Algebras
In this paper we extend a well-known theorem of M.~Scheunert on skew-symmetric bicharacters of groups to the case of skew-symmetric bicharacters on arbitrary cocommutative Hopf algebras over a field of characteristic not 2. We also classify polycharacters on (restricted) enveloping algebras and bicharacters on divided power algebras.

Categories:16W30, 16W55

61. CMB 2001 (vol 44 pp. 27)

Goodaire, Edgar G.; Milies, César Polcino
Normal Subloops in the Integral Loop Ring of an $\RA$ Loop
We show that an $\RA$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\RA$ loop can be normal in its unit loop. Over fields, this can never happen.

Categories:20N05, 17D05, 16S34, 16U60

62. CMB 2000 (vol 43 pp. 413)

Chatters, A. W.
Non-Isomorphic Maximal Orders with Isomorphic Matrix Rings
We construct a countably infinite family of pairwise non-isomorphic maximal ${\mathbb Q}[X]$-orders such that the full $2$ by $2$ matrix rings over these orders are all isomorphic.

Categories:16S50, 16H05, 16N60

63. CMB 2000 (vol 43 pp. 3)

Adin, Ron; Blanc, David
Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedron
Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

64. CMB 2000 (vol 43 pp. 100)

Okon, James S.; Vicknair, J. Paul
A Gorenstein Ring with Larger Dilworth Number than Sperner Number
A counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension $3$ with larger Dilworth number than Sperner number. The Dilworth number of $A[Z/pZ\oplus Z/pZ]$ is computed when $A$ is an unramified principal Artin local ring.

Categories:13E15, 16S34

65. CMB 2000 (vol 43 pp. 79)

König, Steffen
Cyclotomic Schur Algebras and Blocks of Cyclic Defect
An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

Categories:20G05, 20C20, 16G30, 17B37, 57M25

66. CMB 2000 (vol 43 pp. 60)

Farkas, Daniel R.; Linnell, Peter A.
Trivial Units in Group Rings
Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U = G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

Keywords:units, trace, finite conjugate subgroup
Categories:16S34, 16U60

67. CMB 1999 (vol 42 pp. 298)

Jespers, Eric; Okniński, Jan
Semigroup Algebras and Maximal Orders
We describe contracted semigroup algebras of Malcev nilpotent semigroups that are prime Noetherian maximal orders.

Categories:16S36, 16H05, 20M25

68. CMB 1999 (vol 42 pp. 401)

Swain, Gordon A.; Blau, Philip S.
Lie Derivations in Prime Rings With Involution
Let $R$ be a non-GPI prime ring with involution and characteristic $\neq 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$ into itself. Then $\delta=\rho+\epsilon$ where $\epsilon$ is an additive map into the skew elements of the extended centroid of $R$ which is zero on $[K,K]$, and $\rho$ can be extended to an ordinary derivation of $\langle K\rangle$ into $RC$, the central closure.

Categories:16W10, 16N60, 16W25

69. CMB 1999 (vol 42 pp. 371)

Marubayashi, H.; Ueda, A.
Prime and Primary Ideals in a Prüfer Order in a Simple Artinian Ring with Finite Dimension over its Center
Let $Q$ be a simple Artinian ring with finite dimension over its center. An order $R$ in $Q$ is said to be {\it Pr\"ufer\/} if any one-sided $R$-ideal is a progenerator. We study prime and primary ideals of a Pr\"ufer order under the condition that the center is Pr\"ufer. Also we characterize branched and unbranched prime ideals of a Pr\"ufer order.

Categories:16H05, 16L30

70. CMB 1999 (vol 42 pp. 174)

Ferrero, Miguel; Sant'Ana, Alveri
Rings With Comparability
The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

Categories:16U99, 16P40, 16D14, 16N60

71. CMB 1998 (vol 41 pp. 481)

Parmenter, M. M.; Spiegel, E.; Stewart, P. N.
The periodic radical of group rings and incidence algebras
Let $R$ be a ring with $1$ and $P(R)$ the periodic radical of $R$. We obtain necessary and sufficient conditions for $P(\RG) = 0$ when $\RG$ is the group ring of an $\FC$ group $G$ and $R$ is commutative. We also obtain a complete description of $P\bigl(I (X, R)\bigr)$ when $I(X,R)$ is the incidence algebra of a locally finite partially ordered set $X$ and $R$ is commutative.

Categories:16S34, 16S99, 16N99

72. CMB 1998 (vol 41 pp. 452)

Brešar, Matej; Martindale, W. S.; Miers, C. Robert
Dependent automorphisms in prime rings
For each $n\geq 4$ we construct a class of examples of a minimal $C$-dependent set of $n$ automorphisms of a prime ring $R$, where $C$ is the extended centroid of $R$. For $n=4$ and $n=5$ it is shown that the preceding examples are completely general, whereas for $n=6$ an example is given which fails to enjoy any of the nice properties of the above example.

Categories:16N60, 16W20

73. CMB 1998 (vol 41 pp. 359)

Van Oystaeyen, Fred; Zhang, Yinhuo
Embedding the Hopf automorphism group into the Brauer group
Let $H$ be a faithfully projective Hopf algebra over a commutative ring $k$. In \cite{CVZ1, CVZ2} we defined the Brauer group $\BQ(k,H)$ of $H$ and an homomorphism $\pi$ from Hopf automorphism group $\Aut_{\Hopf}(H)$ to $\BQ(k,H)$. In this paper, we show that the morphism $\pi$ can be embedded into an exact sequence.

Categories:16W30, 13A20

74. CMB 1998 (vol 41 pp. 261)

Barthwal, S.; Jhingan, S.; Kanwar, P.
A simple ring over which proper cyclics are continuous is a $\PCI$-ring
It is shown that simple rings over which proper cyclic right modules are continuous coincide with simple right $\PCI$-rings, introduced by Faith.

Keywords:Simple rings, $\PCI$-rings, $\PCQI$-rings, continuous modules,, quasi-continuous modules
Categories:16D50, 16D70

75. CMB 1998 (vol 41 pp. 81)

Lanski, Charles
The cardinality of the center of a $\PI$ ring
The main result shows that if $R$ is a semiprime ring satisfying a polynomial identity, and if $Z(R)$ is the center of $R$, then $\card R \leq 2^{\card Z(R)}$. Examples show that this bound can be achieved, and that the inequality fails to hold for rings which are not semiprime.

Categories:16R20, 16N60, 16R99, 16U50
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