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A graph associated to proper non-small ideals of a commutative ring

S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring R , denoted by G ( R ) , is a graph with all non-small proper ideals of R as vertices and two distinct vertices I and J are adjacent if and only if I J is not small in R . In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...

A note on finitely generated ideal-simple commutative semirings

Vítězslav Kala, Tomáš Kepka (2008)

Commentationes Mathematicae Universitatis Carolinae

Many infinite finitely generated ideal-simple commutative semirings are additively idempotent. It is not clear whether this is true in general. However, to solve the problem, one can restrict oneself only to parasemifields.

An ideal-based zero-divisor graph of direct products of commutative rings

S. Ebrahimi Atani, M. Shajari Kohan, Z. Ebrahimi Sarvandi (2014)

Discussiones Mathematicae - General Algebra and Applications

In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.

Bi-ideals in k-regular and intra k-regular semirings

Anjan K. Bhuniya, Kanchan Jana (2011)

Discussiones Mathematicae - General Algebra and Applications

Here we introduce the k-bi-ideals in semirings and the intra k-regular semirings. An intra k-regular semiring S is a semiring whose additive reduct is a semilattice and for each a ∈ S there exists x ∈ S such that a + xa²x = xa²x. Also it is a semiring in which every k-ideal is semiprime. Our aim in this article is to characterize both the k-regular semirings and intra k-regular semirings using of k-bi-ideals.

Codimension B-W d’un idéal à droite non nul de A 1 ( )

Mathias Konan Kouakou (2005)

Bulletin de la Société Mathématique de France

Soit A 1 ( ) la première algèbre de Weyl sur . La codimension B-W d’un idéal à droite non nul I de A 1 ( ) a été introduite par Yuri Berest et George Wilson. Nous montrons d’une part que cette codimension est invariante par la relation de Stafford : si x Q 1 = Frac ( A 1 ( ) ) , le corps de fractions de A 1 ( ) , et si σ Aut ( A 1 ( ) ) , le groupe des -automorphismes de A 1 ( ) , sont tels que J = x σ ( I ) soit un idéal à droite de A 1 ( ) , alors codim I = codim x σ ( I ) . Nous relions d’autre part la codimension d’un idéal I à la codimension de Gail Letzter-Makar Limanov, de End ( I ) , l’anneau des endomorphismes...

Diagonal reductions of matrices over exchange ideals

Huanyin Chen (2006)

Czechoslovak Mathematical Journal

In this paper, we introduce related comparability for exchange ideals. Let I be an exchange ideal of a ring R . If I satisfies related comparability, then for any regular matrix A M n ( I ) , there exist left invertible U 1 , U 2 M n ( R ) and right invertible V 1 , V 2 M n ( R ) such that U 1 V 1 A U 2 V 2 = diag ( e 1 , , e n ) for idempotents e 1 , , e n I .

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