51. CMB 2009 (vol 40 pp. 54)
 Kechagias, Nondas E.

A note on $U_n\times U_m$ modular invariants
We consider the rings of invariants $R^G$, where $R$ is the symmetric
algebra of a tensor product between two vector spaces over the field $F_p$
and $G=U_n\times U_m$. A polynomial algebra is constructed and these
invariants provide Chern classes for the modular cohomology of $U_{n+m}$.
Keywords:Invariant theory, cohomology of the unipotent group Category:13F20 

52. CMB 2009 (vol 52 pp. 72)
53. CMB 2008 (vol 51 pp. 439)
 Samei, Karim

On the Maximal Spectrum of Semiprimitive Multiplication Modules
An $R$module $M$ is called a multiplication module if for each
submodule $N$ of $M$, $N=IM$ for some ideal $I$ of $R$. As
defined for a commutative ring $R$, an $R$module $M$ is said to be
semiprimitive if the intersection of maximal submodules of $M$ is
zero. The maximal spectra of a semiprimitive multiplication
module $M$ are studied. The isolated points of $\Max(M)$ are
characterized algebraically. The relationships among the maximal
spectra of $M$, $\Soc(M)$ and $\Ass(M)$ are studied. It is shown
that $\Soc(M)$ is exactly the set of all elements of $M$ which
belongs to every maximal submodule of $M$ except for a finite
number. If $\Max(M)$ is infinite, $\Max(M)$ is a onepoint
compactification of a discrete space if and only if $M$ is Gelfand and for
some maximal submodule $K$, $\Soc(M)$ is the intersection of all
prime submodules of $M$ contained in $K$. When $M$ is a
semiprimitive Gelfand module, we prove that every intersection
of essential submodules of $M$ is an essential submodule if and only if
$\Max(M)$ is an almost discrete space. The set of uniform
submodules of $M$ and the set of minimal submodules of $M$
coincide. $\Ann(\Soc(M))M$ is a summand submodule of $M$ if and only if
$\Max(M)$ is the union of two disjoint open subspaces $A$ and
$N$, where $A$ is almost discrete and $N$ is dense in itself. In
particular, $\Ann(\Soc(M))=\Ann(M)$ if and only if $\Max(M)$ is almost
discrete.
Keywords:multiplication module, semiprimitive module, Gelfand module, Zariski topolog Category:13C13 

54. CMB 2008 (vol 51 pp. 406)
 Mimouni, Abdeslam

Condensed and Strongly Condensed Domains
This paper deals with the concepts of condensed and strongly condensed
domains. By definition, an integral domain $R$ is condensed
(resp. strongly condensed) if each pair of ideals $I$ and $J$ of $R$,
$IJ=\{ab/a \in I, b \in J\}$ (resp. $IJ=aJ$ for some $a \in I$ or
$IJ=Ib$ for some $b \in J$). More precisely, we investigate the
ideal theory of condensed and strongly condensed domains in
Noetherianlike settings, especially Mori and strong Mori domains and
the transfer of these concepts to pullbacks.
Categories:13G05, 13A15, 13F05, 13E05 

55. CMB 2007 (vol 50 pp. 598)
56. CMB 2005 (vol 48 pp. 275)
 Smith, Patrick F.

Krull Dimension of Injective Modules Over Commutative Noetherian Rings
Let $R$ be a commutative Noetherian
integral domain with field of fractions $Q$. Generalizing a
fortyyearold theorem of E. Matlis, we prove that the $R$module
$Q/R$ (or $Q$) has Krull dimension if and only if $R$ is semilocal
and onedimensional. Moreover, if $X$ is an injective module over
a commutative Noetherian ring such that $X$ has Krull dimension,
then the Krull dimension of $X$ is at most $1$.
Categories:13E05, 16D50, 16P60 

57. CMB 2003 (vol 46 pp. 304)
 Traves, William N.

Localization of the HasseSchmidt Algebra
The behaviour of the HasseSchmidt algebra of higher derivations under
localization is studied using Andr\'e cohomology. Elementary
techniques are used to describe the HasseSchmidt derivations on
certain monomial rings in the nonmodular case. The localization
conjecture is then verified for all monomial rings.
Categories:13D03, 13N10 

58. CMB 2003 (vol 46 pp. 3)
 Anderson, D. D.; Dumitrescu, Tiberiu

Condensed Domains
An integral domain $D$ with identity is condensed (resp., strongly
condensed) if for each pair of ideals $I$, $J$ of $D$, $IJ=\{ij; i\in I,
j\in J\}$ (resp., $IJ=iJ$ for some $i\in I$ or $IJ =Ij$ for some
$j\in J$). We show that for a Noetherian domain $D$, $D$ is condensed
if and only if $\Pic(D)=0$ and $D$ is locally condensed, while a local
domain is strongly condensed if and only if it has the twogenerator
property. An integrally closed domain $D$ is strongly condensed if and
only if $D$ is a B\'{e}zout generalized Dedekind domain with at most one
maximal ideal of height greater than one. We give a number of
equivalencies for a local domain with finite integral closure to be
strongly condensed. Finally, we show that for a field extension
$k\subseteq K$, the domain $D=k+XK[[X]]$ is condensed if and only if
$[K:k]\leq 2$ or $[K:k]=3$ and each degreetwo polynomial in $k[X]$
splits over $k$, while $D$ is strongly condensed if and only if $[K:k]
\leq 2$.
Categories:13A15, 13B22 

59. CMB 2002 (vol 45 pp. 272)
 Neusel, Mara D.

The Transfer in the Invariant Theory of Modular Permutation Representations II
In this note we show that the image of the transfer for permutation
representations of finite groups is generated by the transfers of
special monomials. This leads to a description of the image of the
transfer of the alternating groups. We also determine the height of
these ideals.
Keywords:polynomial invariants of finite groups, permutation representation, transfer Category:13A50 

60. CMB 2002 (vol 45 pp. 119)
61. CMB 2000 (vol 43 pp. 362)
 Kim, Hwankoo

Examples of HalfFactorial Domains
In this paper, we determine some sufficient conditions for an $A +
XB[X]$ domain to be an HFD. As a consequence we give new examples
of HFDs of the type $A + XB[X]$.
Keywords:atomic domain, HFD Categories:13A05, 13B30, 13F15, 13G05 

62. CMB 2000 (vol 43 pp. 312)
 Dobbs, David E.

On the Prime Ideals in a Commutative Ring
If $n$ and $m$ are positive integers, necessary and sufficient
conditions are given for the existence of a finite commutative ring $R$
with exactly $n$ elements and exactly $m$ prime ideals. Next,
assuming the Axiom of Choice, it is proved that if $R$ is a
commutative ring and $T$ is a commutative $R$algebra which is
generated by a set $I$, then each chain of prime ideals of $T$ lying
over the same prime ideal of $R$ has at most $2^{I}$ elements. A
polynomial ring example shows that the preceding result is
bestpossible.
Categories:13C15, 13B25, 04A10, 14A05, 13M05 

63. CMB 2000 (vol 43 pp. 126)
64. CMB 2000 (vol 43 pp. 100)
65. CMB 1999 (vol 42 pp. 231)
 Rush, David E.

Generating Ideals in Rings of IntegerValued Polynomials
Let $R$ be a onedimensional locally analytically irreducible
Noetherian domain with finite residue fields. In this note it is
shown that if $I$ is a finitely generated ideal of the ring
$\Int(R)$ of integervalued polynomials such that for each $x \in
R$ the ideal $I(x) =\{f(x) \mid f \in I\}$ is strongly
$n$generated, $n \geq 2$, then $I$ is $n$generated, and some
variations of this result.
Categories:13B25, 13F20, 13F05 

66. CMB 1999 (vol 42 pp. 155)
 Campbell, H. E. A.; Geramita, A. V.; Hughes, I. P.; Shank, R. J.; Wehlau, D. L.

NonCohenMacaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
This paper contains two essentially independent results in the
invariant theory of finite groups. First we prove that, for any
faithful representation of a nontrivial $p$group over a field of
characteristic $p$, the ring of vector invariants of $m$ copies of
that representation is not \comac\ for $m\geq 3$. In the second
section of the paper we use Poincar\'e series methods to produce upper
bounds for the degrees of the generators for the ring of invariants as
long as that ring is Gorenstein. We prove that, for a finite
nontrivial group $G$ and a faithful representation of dimension $n$
with $n>1$, if the ring of invariants is Gorenstein then the ring is
generated in degrees less than or equal to $n(G1)$. If the ring of
invariants is a hypersurface, the upper bound can be improved to $G$.
Category:13A50 

67. CMB 1999 (vol 42 pp. 125)
 Smith, Larry

Modular Vector Invariants of Cyclic Permutation Representations
Vector invariants of finite groups (see the introduction for an
explanation of the terminology) have often been used to illustrate the
difficulties of invariant theory in the modular case: see,
\eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB},
\cite{poly} and \cite{survey}. It is therefore all the more
surprising that the {\it unpleasant} properties of these invariants
may be derived from two unexpected, and remarkable, {\it nice}
properties: namely for vector permutation invariants of the cyclic
group $\mathbb{Z}/p$ of prime order in characteristic $p$ the
image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon
\mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal,
and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im
(\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern
classes of the action.
Keywords:polynomial invariants of finite groups Category:13A50 

68. CMB 1998 (vol 41 pp. 359)
69. CMB 1998 (vol 41 pp. 28)
70. CMB 1998 (vol 41 pp. 3)
 Anderson, David F.; Dobbs, David E.

Root closure in Integral Domains, III
{If A is a subring of a commutative ring B and if n
is a positive integer, a number of sufficient conditions are given for
``A[[X]]is nroot closed in B[[X]]'' to be equivalent to ``A is nroot
closed in B.'' In addition, it is shown that if S is a multiplicative
submonoid of the positive integers ${\bbd P}$ which is generated by
primes, then there exists a onedimensional quasilocal integral domain
A (resp., a von Neumann regular ring A) such that $S = \{ n \in {\bbd P}\mid
A$ is $n$root closed$\}$ (resp., $S = \{n \in {\bbd P}\mid A[[X]]$
is $n$rootclosed$\}$).
Categories:13G05, 13F25, 13C15, 13F45, 13B99, 12D99 
