scholarly journals Independence complexes of comaximal graphs of commutative rings with identity

2015 ◽  
Vol 98 (112) ◽  
pp. 109-117
Author(s):  
Nela Milosevic
Keyword(s):  
Commutative Rings ◽  
Homotopy Equivalent ◽  
Comaximal Graph ◽  
Independence Complex

We study topology of the independence complexes of comaximal (hyper)graphs of commutative rings with identity. We show that the independence complex of comaximal hypergraph is contractible or homotopy equivalent to a sphere, and that the independence complex of comaximal graph is almost always contractible.

scholarly journals A Note on Independence Complexes of Chordal Graphs and Dismantling

10.37236/5571 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Michał Adamaszek
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Homotopy Equivalent ◽  
Tree Models ◽  
Independence Complex

We show that the independence complex of a chordal graph is contractible if and only if this complex is dismantlable (strong collapsible) and it is homotopy equivalent to a sphere if and only if its core is a cross-polytopal sphere. The proof uses the properties of tree models of chordal graphs.

scholarly journals A Note on Comaximal Graph of Non-commutative Rings

2011 ◽  
Vol 16 (2) ◽  
pp. 303-307 ◽  
Author(s):  
Saieed Akbari ◽  
Mohammad Habibi ◽  
Ali Majidinya ◽  
Raoofe Manaviyat
Keyword(s):  
Commutative Rings ◽  
Comaximal Graph

Comaximal graph of amalgamated algebras along an ideal

2021 ◽  
Author(s):  
Hanieh Shoar ◽  
Maryam Salimi ◽  
Abolfazl Tehranian ◽  
Hamid Rasouli ◽  
Elham Tavasoli
Keyword(s):  
Graph Theory ◽  
Jacobson Radical ◽  
Commutative Rings ◽  
Ring Homomorphism ◽  
Algebraic Properties ◽  
Comaximal Graph

Let [Formula: see text] and [Formula: see text] be commutative rings with identity, [Formula: see text] be an ideal of [Formula: see text], and let [Formula: see text] be a ring homomorphism. The amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text] denoted by [Formula: see text] was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of [Formula: see text] which are transferred to the comaximal graph of [Formula: see text], and also we study some algebraic properties of the ring [Formula: see text] by way of graph theory. The comaximal graph of [Formula: see text], [Formula: see text], was introduced by Sharma and Bhatwadekar in 1995. The vertices of [Formula: see text] are all elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. Let [Formula: see text] be the subgraph of [Formula: see text] generated by non-unit elements, and let [Formula: see text] be the Jacobson radical of [Formula: see text]. It is shown that the diameter of the graph [Formula: see text] is equal to the diameter of the graph [Formula: see text], and the girth of the graph [Formula: see text] is equal to the girth of the graph [Formula: see text], provided some special conditions.

scholarly journals Comaximal graph of commutative rings

2008 ◽  
Vol 319 (4) ◽  
pp. 1801-1808 ◽  
Author(s):  
Hamid Reza Maimani ◽  
Maryam Salimi ◽  
Asiyeh Sattari ◽  
Siamak Yassemi
Keyword(s):  
Commutative Rings ◽  
Comaximal Graph

ON THE COZERO-DIVISOR GRAPHS AND COMAXIMAL GRAPHS OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250173 ◽  
Author(s):  
MOJGAN AFKHAMI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Comaximal Graph ◽  
Vertex Set

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertices in W*(R), which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ bR and b ∉ aR. Also the comaximal graph is a graph with vertices all elements of R and two distinct vertices a and b are adjacent if and only if aR + bR = R. We denote the subgraph of the comaximal graph with vertex-set W*(R), by Γ2(R). In this note, we study the relations between two graphs Γ′(R) and Γ2(R). Also, we investigate the cozero-divisor graphs of idealizations of commutative rings.

scholarly journals ON THE STRUCTURE OF COMAXIMAL GRAPHS OF COMMUTATIVE RINGS WITH IDENTITY

2010 ◽  
Vol 83 (1) ◽  
pp. 11-21 ◽  
Author(s):  
SLAVKO M. MOCONJA ◽  
ZORAN Z. PETROVIĆ
Keyword(s):  
Commutative Rings ◽  
Comaximal Graph

AbstractIn this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms of comaximal graphs and the rings in question. In addition, we investigate the relation between the comaximal graph of a ring and its subrings of a certain type.

GENERALIZED OF PRIMARY AVOIDANCE THEOREM FOR COMMUTATIVE RINGS

2018 ◽  
Vol 1 (21) ◽  
pp. 415-438
Author(s):  
Amer Shamil Abdulrhman
Keyword(s):  
Commutative Rings ◽  
Primary Ideals

In this paper we study covering ideals by Cosets of primary ideals and we get a generalized the primary avoidance theorem in the rings which it has been

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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