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BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY

Assume that $(R;m)$ is a local Noetherian ring and $I$ is an ideal of $R$. In this paper

we introduce a new class of $R$-modules denoted by weakly finite modules that is a generalization

of finitely generated modules and containing the class of Big Cohen-Macaulay modules

and $I$-cofinite modules. We improve the non-vanishing theorem due to Grothendieck

for weakly finite modules. Finally we define the notion $depth_R (M)$ and we prove that if $M$ is

a weakly finite $R$-module and $H^i_

m(M) = 0$ for some i, then $depth_R(M)leq ileq dim(M)$.

- Local Cohomology Modules
- Grothendieck’s Non-Vanishing Theorem
- Big Cohen-Macaulay modules

Mathematical Research Letters

Let $G$ be an abelian group and $S$ be a $G$-graded a Noetherian algebra over a commutative ring $Asubseteq S_0$. Let $I_1, dots, I_s$ be $G$-homogeneous ideals in $S$, and let $M$ be a finitely generated $G$-graded $S$-module. We show that the shape of nonzero $G$-graded Betti numbers of $MI_1^{t_1} dots I_s^{t_s}$ exhibit an eventual linear behavior as the $t_i$s get large.

- Asymptotic Linearity
- Castelnuovo-Mumford Regularity
- Multigraded Regularity,
- Betti Number

Journal of Algebra and Its Applications Vol. 11, No. 1 (2012) 1250013 (9 pages)

Let $R$ be a commutative Noetherian ring and let $I$ be an ideal of $R$. In this paper,

we study the amalgamated duplication ring $Rbowtie I$ which is introduced by D’Anna and

Fontana. It is shown that if $R$ satisfies Serre’s condition $(S_n)$ and $I_fp$ is a maximal Cohen–

Macaulay $R_fp$-module for every $fp in Spec (R)R, then $Rbowtie I$ satisfies Serre’s condition

$(S_n)$. Moreover if $Rbowtie I$ satisfies Serre’s condition $(S_n)$, then so does $R$. This gives a

generalization of the same result for Cohen–Macaulay rings in [D’Anna, A construction

of Gorenstein rings, J. Algebra 306 (2006) 507–519]. In addition it is shown that if $R$ is

a local ring and $Ann R(I) = 0$, then $Rbowtie I$ is quasi-Gorenstein if and only if $hat R$

satisfies

Serre’s condition $(S_2)$ and $I$ is a canonical ideal of $R$. This result improves the result of

D’Anna which is corrected by Shapiro and states that if $R$ is a Cohen–Macaulay local

ring, then $Rbowtie I$ is Gorenstein if and only if the canonical ideal of $R$ exists and is

isomorphic to $I$, provided $Ann R(I) = 0$.

- Quasi-Gorenstein
- Amalgamated Duplication