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Search: MSC category 13E05 ( Noetherian rings and modules )

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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 2012 (vol 56 pp. 683)

Nikseresht, A.; Azizi, A.
Envelope Dimension of Modules and the Simplified Radical Formula
We introduce and investigate the notion of envelope dimension of commutative rings and modules over them. In particular, we show that the envelope dimension of a ring, $R$, is equal to that of the $R$-module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a ring is no more than its envelope dimension and characterize Noetherian rings for which these two dimensions are equal. Moreover we generalize and study the concept of simplified radical formula for modules, which we defined in an earlier paper.

Keywords:envelope dimension, simplified radical formula, prime submodule
Categories:13A99, 13C99, 13C13, 13E05

3. CMB 2011 (vol 55 pp. 378)

Oman, Greg; Salminen, Adam
On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality
Let $R$ be a commutative ring with identity, and let $M$ be a unitary module over $R$. We call $M$ H-smaller (HS for short) if and only if $M$ is infinite and $|M/N|<|M|$ for every nonzero submodule $N$ of $M$. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose $M$ is faithful over $R$, $R$ is a domain (we will show that we can restrict to this case without loss of generality), and $K$ is the quotient field of $R$. If $M$ is HS over $R$, then $R$ is HS as a module over itself, $R\subseteq M\subseteq K$, and there exists a generating set $S$ for $M$ over $R$ with $|S|<|R|$. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.

Keywords:Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, Jónsson module
Categories:13A99, 13C05, 13E05, 03E50

4. 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

5. 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

6. 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

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