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51. 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

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

60. 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

61. 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

62. 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

63. 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

64. 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

65. 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

66. 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

67. 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

68. 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

69. 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

70. CMB 1997 (vol 40 pp. 221)

Rada, Juan; Saorín, Manuel
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

71. 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

72. CMB 1997 (vol 40 pp. 103)

Riley, David M.; Tasić, Vladimir
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

73. CMB 1997 (vol 40 pp. 47)

Hartl, Manfred
A universal coefficient decomposition for subgroups induced by submodules of group algebras
Dimension subgroups and Lie dimension subgroups are known to satisfy a `universal coefficient decomposition', {\it i.e.} their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the `universal' coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary $N$-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.

Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial ideals
Categories:20C07, 16A27
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