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51. 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

52. CMB 2002 (vol 45 pp. 388)

Gille, Philippe
 AlgÃ¨bres simples centrales de degrÃ© 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:algÃ¨bres simples centrales, cohomologie galoisienneCategories:16S35, 12G05, 20G15

53. 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

54. 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

55. 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

56. 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

57. 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 subgroupCategories:16S34, 16U60

58. 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

59. 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

60. CMB 2000 (vol 43 pp. 3)

 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, associahedronCategories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50

61. 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

62. 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

63. CMB 1999 (vol 42 pp. 371)

Marubayashi, H.; Ueda, A.
 Prime and Primary Ideals in a PrÃ¼fer 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

64. 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

65. 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

66. 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

67. 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

68. 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 modulesCategories:16D50, 16D70

69. 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

70. CMB 1998 (vol 41 pp. 118)

Valenti, Angela
 On permanental identities of symmetric and skew-symmetric matrices in characteristic \lowercase{$p$} Let $M_n(F)$ be the algebra of $n \times n$ matrices over a field $F$ of characteristic $p>2$ and let $\ast$ be an involution on $M_n(F)$. If $s_1, \ldots, s_r$ are symmetric variables we determine the smallest $r$ such that the polynomial $$P_{r}(s_1, \ldots, s_{r}) = \sum_{\sigma \in {\cal S}_r}s_{\sigma(1)}\cdots s_{\sigma(r)}$$ is a $\ast$-polynomial identity of $M_n(F)$ under either the symplectic or the transpose involution. We also prove an analogous result for the polynomial $$C_r(k_1, \ldots, k_r, k'_1, \ldots, k'_r) = \sum_ {\sigma, \tau \in {\cal S}_r}k_{\sigma(1)}k'_{\tau(1)}\cdots k_{\sigma(r)}k'_{\tau(r)}$$ where $k_1, \ldots, k_r, k'_1, \ldots, k'_r$ are skew variables under the transpose involution. Category:16R50

71. CMB 1998 (vol 41 pp. 109)

Tahara, Ken-Ichi; Vermani, L. R.; Razdan, Atul
 On generalized third dimension subgroups Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M.~Hartl identified the subgroup $G \cap(1+\triangle^3(G)+\triangle(G)\triangle(H))$ of $G$. In this note we give an independent proof of the result of Hartl, and we identify two subgroups $G\cap(1+\triangle(H)\triangle(G)\triangle(H)+\triangle([H,G])\triangle(H))$, $G\cap(1+\triangle^2(G)\triangle(H)+\triangle(K)\triangle(H))$ of $G$ for some subgroup $K$ of $G$ containing $[H,G]$. Categories:20C07, 16S34

72. CMB 1998 (vol 41 pp. 79)

Kelarev, A. V.
 An answer to a question of Kegel on sums of rings We construct a ring $R$ which is a sum of two subrings $A$ and $B$ such that the Levitzki radical of $R$ does not contain any of the hyperannihilators of $A$ and $B$. This answers an open question asked by Kegel in 1964. Categories:16N40, 16N60

73. CMB 1997 (vol 40 pp. 221)

 On semiregular rings whose finitely generated modules embed in free modules We consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question. Categories:16D10, 16L60, 16N20

74. CMB 1997 (vol 40 pp. 198)

Meldrum, J. D. P.; Meyer, J. H.
 The ${\cal J}_0$-radical of a matrix nearring can be intermediate An example is constructed to show that the ${\cal J}_0$-radical of a matrix nearring can be an intermediate ideal. This solves a conjecture put forward in [1]. Categories:16Y30, 16S50, 16D25

75. CMB 1997 (vol 40 pp. 103)

 The transfer of a commutator law from a nil-ring to its adjoint group For every field $F$ of characteristic $p\geq 0$, we construct an example of a finite dimensional nilpotent $F$-algebra $R$ whose adjoint group $A(R)$ is not centre-by-metabelian, in spite of the fact that $R$ is Lie centre-by-metabelian and satisfies the identities $x^{2p}=0$ when $p>2$ and $x^8=0$ when $p=2$. The existence of such algebras answers a question raised by A.~E.~Zalesskii, and is in contrast to positive results obtained by Krasilnikov, Sharma and Srivastava for Lie metabelian rings and by Smirnov for the class Lie centre-by-metabelian nil-algebras of exponent 4 over a field of characteristic 2 of cardinality at least 4. Categories:16U60, 17B60