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26. CMB 2010 (vol 53 pp. 667)

Khashyarmanesh, Kazem
 On the Endomorphism Rings of Local Cohomology Modules Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n:=\operatorname{grade}_R\mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))\cong \operatorname{Ext}_R^n(H^n_\mathfrak{a}(R),R)$. We also prove that, for a nonnegative integer $n$ such that $H^i_\mathfrak{a}(R)=0$ for every $i\neq n$, if $\operatorname{Ext}_R^i(R_z,R)=0$ for all $i >0$ and $z \in \mathfrak{a}$, then $\operatorname{End}_R(H^n_\mathfrak{a}(R))$ is a homomorphic image of $R$, where $R_z$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{z^j \mid j \geqslant 0 \}$ of $R$. Moreover, if $\operatorname{Hom}_R(R_z,R)=0$ for all $z \in \mathfrak{a}$, then $\mu_{H^n_\mathfrak{a}(R)}$ is an isomorphism, where $\mu_{H^n_\mathfrak{a}(R)}$ is the canonical ring homomorphism $R \rightarrow \operatorname{End}_R(H^n_\mathfrak{a}(R))$. Keywords:local cohomology module, endomorphism ring, Matlis dual functor, filter regular sequenceCategories:13D45, 13D07, 13D25

27. CMB 2010 (vol 53 pp. 404)

Broer, Abraham
 Invariant Theory of Abelian Transvection Groups Let $G$ be a finite group acting linearly on the vector space $V$ over a field of arbitrary characteristic. The action is called \emph{coregular} if the invariant ring is generated by algebraically independent homogeneous invariants, and the \emph{direct summand property} holds if there is a surjective $k[V]^G$-linear map $\pi\colon k[V]\to k[V]^G$. The following Chevalley--Shephard--Todd type theorem is proved. Suppose $G$ is abelian. Then the action is coregular if and only if $G$ is generated by pseudo-reflections and the direct summand property holds. Category:13A50

28. CMB 2009 (vol 53 pp. 77)

Finston, David; Maubach, Stefan
 Constructing (Almost) Rigid Rings and a UFD Having Infinitely Generated Derksen and Makar-Limanov Invariants An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem. Categories:14R20, 13A50, 13N15

29. CMB 2009 (vol 53 pp. 247)

Etingof, P.; Malcolmson, P.; Okoh, F.
 Root Extensions and Factorization in Affine Domains An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element a in R, the ascending chain of non-associate irreducible divisors in R of $a^{n}$ stabilizes on a finite set as n ranges over the positive integers, while R is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension S of R is a \emph{root extension} or \emph{radical extension} if for each s in S, there exists a natural number $n(s)$ with $s^{n(s)}$ in R. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $(R,S)$ is governed by the relative sizes of the unit groups $\operatorname{U}(R)$ and $\operatorname{U}(S)$ and whether S is a root extension of R. The following results are deduced from these considerations. An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let R be a Noetherian domain with integral closure S. Suppose the conductor of S into R is non-zero. Then R is IDPF if and only if S is a root extension of R and $\operatorname{U}(S)/\operatorname{U}(R)$ is finite. Categories:13F15, 14A25

30. CMB 2009 (vol 52 pp. 535)

Daigle, Daniel; Kaliman, Shulim
 A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$ We strengthen certain results concerning actions of $(\Comp,+)$ on $\Comp^{3}$ and embeddings of $\Comp^{2}$ in $\Comp^{3}$, and show that these results are in fact valid over any field of characteristic zero. Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine spaceCategories:14R10, 14R20, 14R25, 13N15

31. CMB 2009 (vol 52 pp. 72)

Duncan, Alexander; LeBlanc, Michael; Wehlau, David L.
 A SAGBI Basis For $\mathbb F[V_2\oplus V_2\oplus V_3]^{C_p}$ Let $C_p$ denote the cyclic group of order $p$, where $p \geq 3$ is prime. We denote by $V_n$ the indecomposable $n$ dimensional representation of $C_p$ over a field $\mathbb F$ of characteristic $p$. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants $\mathbb F[V_2 \oplus V_2 \oplus V_3]^{C_p}$. Category:13A50

32. 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 Noetherian-like settings, especially Mori and strong Mori domains and the transfer of these concepts to pullbacks. Categories:13G05, 13A15, 13F05, 13E05

33. 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 one-point 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 topologCategory:13C13

34. CMB 2007 (vol 50 pp. 598)

Lorestani, Keivan Borna; Sahandi, Parviz; Yassemi, Siamak
 Artinian Local Cohomology Modules Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\H^i_\fa(M)$ is finitely generated for all $i Keywords:local cohomology module, Artinian module, reflexive moduleCategories:13D45, 13E10, 13C05 35. CMB 2005 (vol 48 pp. 275) Smith, Patrick F.  Krull Dimension of Injective Modules Over Commutative Noetherian Rings Let$R$be a commutative Noetherian integral domain with field of fractions$Q$. Generalizing a forty-year-old theorem of E. Matlis, we prove that the$R$-module$Q/R$(or$Q$) has Krull dimension if and only if$R$is semilocal and one-dimensional. Moreover, if$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 36. CMB 2003 (vol 46 pp. 304) Traves, William N.  Localization of the Hasse-Schmidt Algebra The behaviour of the Hasse-Schmidt algebra of higher derivations under localization is studied using Andr\'e cohomology. Elementary techniques are used to describe the Hasse-Schmidt derivations on certain monomial rings in the nonmodular case. The localization conjecture is then verified for all monomial rings. Categories:13D03, 13N10 37. 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 two-generator 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 degree-two polynomial in$k[X]$splits over$k$, while$D$is strongly condensed if and only if$[K:k] \leq 2$. Categories:13A15, 13B22 38. 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, transferCategory:13A50 39. CMB 2002 (vol 45 pp. 119) Marcelo, Agustín; Marcelo, Félix; Rodríguez, César  The Grade Conjecture and the$S_{2}$Condition Sufficient conditions are given in order to prove the lowest unknown case of the grade conjecture. The proof combines vanishing results of local cohomology and the$S_{2}$condition. Categories:13D22, 13D45, 13D25, 13C15 40. CMB 2000 (vol 43 pp. 362) Kim, Hwankoo  Examples of Half-Factorial 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, HFDCategories:13A05, 13B30, 13F15, 13G05 41. 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 best-possible. Categories:13C15, 13B25, 04A10, 14A05, 13M05 42. CMB 2000 (vol 43 pp. 126) Soto, José J. M.  Sur l'annulation de certains modules de cohomologie d'AndrÃ©-Quillen Soient$A$un anneau noeth\'erien,$B$un anneau r\'egulier essentiellement de type fini sur$A$. Si la cohomologie d'Andr\'e-Quillen$H^q (A,B,B) = 0$pour tout$q \geq 2$alors$A$est un anneau r\'egulier. Category:13D03 43. 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 44. CMB 1999 (vol 42 pp. 231) Rush, David E.  Generating Ideals in Rings of Integer-Valued Polynomials Let$R$be a one-dimensional 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 integer-valued 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 45. CMB 1999 (vol 42 pp. 155) Campbell, H. E. A.; Geramita, A. V.; Hughes, I. P.; Shank, R. J.; Wehlau, D. L.  Non--Cohen-Macaulay 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 non-trivial$p$-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 non-trivial 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$n(|G|-1)$. If the ring of invariants is a hypersurface, the upper bound can be improved to$|G|$. Category:13A50 46. 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 groupsCategory:13A50 47. 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 48. 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 n-root closed in B[[X]]'' to be equivalent to A is n-root 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 one-dimensional 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 49. CMB 1998 (vol 41 pp. 28) Félix, Yves; Murillo, Aniceto  Gorenstein graded algebras and the evaluation map We consider graded connected Gorenstein algebras with respect to the evaluation map$\ev_G = \Ext_G(k,\varepsilon )=:: \Ext_G(k,G) \longrightarrow \Ext_G(k,k)$. We prove that if$\ev_G \neq 0$, then the global dimension of$G$is finite. Categories:55P35, 13C11 50. CMB 1997 (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 groupCategory:13F20
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