# Commutative rings I

25 Sep 2007-pp 65-83

About: The article was published on 2007-09-25. It has received 425 citations till now. The article focuses on the topics: Commutative ring.

##### Citations

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TL;DR: In this article, the authors introduced and investigated the total graph of R, denoted by T ( Γ ( R ) ), which is the (undirected) graph with all elements of R as vertices.

290 citations

### Cites background from "Commutative rings I"

...General references for ring theory are [10] and [11]....

[...]

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TL;DR: In this paper, strongly Gorenstein projective, injective, and flat modules are studied, which they call strongly GORNEINSTEIN projective and injective.

162 citations

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TL;DR: In this article, a commutative ring R and a proper ideal I ⊂ R were constructed and a new ring denoted by R⋈I was studied.

161 citations

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TL;DR: In this paper, the authors introduce a new general construction, called the amalgamated duplication of a ring $R$ along an ideal module $E$ that they assume to be an ideal in some overring of $R$.

Abstract: We introduce a new general construction, denoted by $R\JoinE$, called the amalgamated duplication of a ring $R$ along an $R$--module $E$, that we assume to be an ideal in some overring of $R$. (Note that, when $E^2 =0$, $R\JoinE$ coincides with the Nagata's idealization $R\ltimes E$.)
After discussing the main properties of the amalgamated duplication $R\JoinE$ in relation with pullback--type constructions, we restrict our investigation to the study of $R\JoinE$ when $E$ is an ideal of $R$.
Special attention is devoted to the ideal-theoretic properties of $R\JoinE$ and to the topological structure of its prime spectrum.

144 citations

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TL;DR: In this paper, a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations with p≥5 was proved.

Abstract: We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL2(Q
p
) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL2(Q
p
) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.

142 citations

### Cites background from "Commutative rings I"

...Since R[1/p] is a field and R is a noetherian integral domain Theorem 146 in [37] implies that R/pR is artinian....

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##### References

More filters

••

TL;DR: In this article, the authors introduced and investigated the total graph of R, denoted by T ( Γ ( R ) ), which is the (undirected) graph with all elements of R as vertices.

290 citations

••

TL;DR: In this article, a commutative ring R and a proper ideal I ⊂ R were constructed and a new ring denoted by R⋈I was studied.

161 citations

•

TL;DR: In this paper, the authors introduce a new general construction, called the amalgamated duplication of a ring $R$ along an ideal module $E$ that they assume to be an ideal in some overring of $R$.

Abstract: We introduce a new general construction, denoted by $R\JoinE$, called the amalgamated duplication of a ring $R$ along an $R$--module $E$, that we assume to be an ideal in some overring of $R$. (Note that, when $E^2 =0$, $R\JoinE$ coincides with the Nagata's idealization $R\ltimes E$.)
After discussing the main properties of the amalgamated duplication $R\JoinE$ in relation with pullback--type constructions, we restrict our investigation to the study of $R\JoinE$ when $E$ is an ideal of $R$.
Special attention is devoted to the ideal-theoretic properties of $R\JoinE$ and to the topological structure of its prime spectrum.

144 citations

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TL;DR: In this paper, the authors studied the problem of nonzero zero-divisor graphs of commutative rings with identity and proved that they satisfy certain divisibility conditions between elements of R or comparability conditions between ideals of R.

Abstract: Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and Nil(R) its ideal of nilpotent elements. The zero-divisor graph of R is Γ(R) = Z(R)\{0}, with distinct vertices x and y adjacent if and only if xy = 0. In this article, we study Γ(R) for rings R with nonzero zero-divisors which satisfy certain divisibility conditions between elements of R or comparability conditions between ideals or prime ideals of R. These rings include chained rings, rings R whose prime ideals contained in Z(R) are linearly ordered, and rings R such that {0} ≠ Nil(R) ⊆ zR for all z ∈ Z(R)\Nil(R).

127 citations

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TL;DR: In this article, it was shown that the annihilating-ideal graph of a commutative ring with a set of ideals with nonzero annihilator is a connected graph.

Abstract: Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is the (undirected) graph with vertices ${\Bbb{A}}(R)^*:={\Bbb{A}}(R)\setminus\{(0)\}$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness conditions of ${\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a domain, then ${\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if $R$ is Noetherian (resp., Artinian). Moreover, the set of vertices of ${\Bbb{AG}}(R)$ and the set of nonzero proper ideals of $R$ have the same cardinality when $R$ is either an Artinian or a decomposable ring. This yields for a ring $R$, ${\Bbb{AG}}(R)$ has $n$ vertices $(n\geq 1)$ if and only if $R$ has only $n$ nonzero proper ideals. Next, we study the connectivity of ${\Bbb{AG}}(R)$. It is shown that ${\Bbb{AG}}(R)$ is a connected graph and $diam(\Bbb{AG})(R)\leq 3$ and if ${\Bbb{AG}}(R)$ contains a cycle, then $gr({\Bbb{AG}}(R))\leq 4$. Also, rings $R$ for which the graph ${\Bbb{AG}}(R)$ is complete or star, are characterized, as well as rings $R$ for which every vertex of ${\Bbb{AG}}(R)$ is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

127 citations