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Search: MSC category 13E10 ( Artinian rings and modules, finite-dimensional algebras )

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1. CMB Online first

Shaveisi, Farzad
Some Results on the Annihilating-Ideal Graphs
The annihilating-ideal graph of a commutative ring $R$, denoted by $\mathbb{AG}(R)$, is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Here, we show that if $R$ is a reduced ring and the independence number of $\mathbb{AG}(R)$ is finite, then the edge chromatic number of $\mathbb{AG}(R)$ equals its maximum degree and this number equals $2^{|{\rm Min}(R)|-1}-1$; also, it is proved that the independence number of $\mathbb{AG}(R)$ equals $2^{|{\rm Min}(R)|-1}$, where ${\rm Min}(R)$ denotes the set of minimal prime ideals of $R$. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a finite graph $\mathbb{AG}(R)$ is not Eulerian, and it is Hamiltonian if and only if $R$ contains no Gorenstain ring as its direct summand.

Keywords:annihilating-ideal graph, independence number, edge chromatic number, bipartite, cycle
Categories:05C15, 05C69, 13E05, 13E10

2. CMB 2016 (vol 59 pp. 271)

Dehghani-Zadeh, Fatemeh
Artinianness of Composed Graded Local Cohomology Modules
Let $R=\bigoplus_{n\geq0}R_{n}$ be a graded Noetherian ring with local base ring $(R_{0}, \mathfrak{m}_{0})$ and let $R_{+}=\bigoplus_{n\gt 0}R_{n}$, $M$ and $N$ be finitely generated graded $R$-modules and $\mathfrak{a}=\mathfrak{a}_{0}+R_{+}$ an ideal of $R$. We show that $H^{j}_{\mathfrak{b}_{0}}(H^{i}_{\mathfrak{a}}(M,N))$ and $H^{i}_{\mathfrak{a}}(M, N)/\mathfrak{b}_{0}H^{i}_{\mathfrak{a}}(M,N)$ are Artinian for some $i^{,}s$ and $j^{,}s$ with a specified property, where $\mathfrak{b}_{o}$ is an ideal of $R_{0}$ such that $\mathfrak{a}_{0}+\mathfrak{b}_{0}$ is an $\mathfrak{m}_{0}$-primary ideal.

Keywords:generalized local cohomology, Artinian, graded module
Categories:13D45, 13E10, 16W50

3. CMB 2011 (vol 55 pp. 81)

Divaani-Aazar, Kamran; Hajikarimi, Alireza
Cofiniteness of Generalized Local Cohomology Modules for One-Dimensional Ideals
Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $\dim R/\mathfrak a\leq 1$, then all generalized local cohomology modules $H^i_{\mathfrak a}(M,N)$ are $\mathfrak a$-cofinite.

Keywords:cofinite modules, generalized local cohomology modules, local cohomology modules
Categories:13D45, 13E05, 13E10

4. CMB 2011 (vol 55 pp. 153)

Mafi, Amir; Saremi, Hero
Artinianness of Certain Graded Local Cohomology Modules
We show that if $R=\bigoplus_{n\in\mathbb{N}_0}R_n$ is a Noetherian homogeneous ring with local base ring $(R_0,\mathfrak{m}_0)$, irrelevant ideal $R_+$, and $M$ a finitely generated graded $R$-module, then $H_{\mathfrak{m}_0R}^j(H_{R_+}^t(M))$ is Artinian for $j=0,1$ where $t=\inf\{i\in{\mathbb{N}_0}: H_{R_+}^i(M)$ is not finitely generated $\}$. Also, we prove that if $\operatorname{cd}(R_+,M)=2$, then for each $i\in\mathbb{N}_0$, $H_{\mathfrak{m}_0R}^i(H_{R_+}^2(M))$ is Artinian if and only if $H_{\mathfrak{m}_0R}^{i+2}(H_{R_+}^1(M))$ is Artinian, where $\operatorname{cd}(R_+,M)$ is the cohomological dimension of $M$ with respect to $R_+$. This improves some results of R. Sazeedeh.

Keywords:graded local cohomology, Artinian modules
Categories:13D45, 13E10

5. CMB 2011 (vol 54 pp. 619)

Dibaei, Mohammad T.; Vahidi, Alireza
Artinian and Non-Artinian Local Cohomology Modules
Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\mathfrak{a}, \mathfrak{b}$, $\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the non-empty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M) $ is not artinian for some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.

Keywords:local cohomology modules, cohomological dimensions, Bass numbers
Categories:13D45, 13E10

6. 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 module
Categories:13D45, 13E10, 13C05

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