Classification of non-local rings with genus two zero-divisor graphs

Asir, T.; Mano, K.

doi:10.1007/s00500-019-04345-0

Cited by 19 publications

(10 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There is no such vertex in Γ (Z 2 × Z 3 × Z 4 ) and hence wrongly included vertex (2, 0, 2) is deleted from the S 2 -embedding of Γ ( Z 2 × Z 3 × Z 4 ) given in Fig. 7 [1].…”

Section: Theorem 1 Let R Be a Non-local Commutative Ring Then The Genus Of The Zero-divisor Graph Is 2 If And Only If R Is Isomorphic To mentioning

confidence: 99%

“…

The purpose of this note is to correct some errors in the article [1]. The notation here will follow that of the original article.ItThis mistake causes a change in the value the genus of Γ (Z 2 × Z 2 × Z 8 ).

…”

mentioning

confidence: 99%

“…The purpose of this note is to correct some errors in the article [1]. The notation here will follow that of the original article.…”

mentioning

confidence: 99%

“…From the above observation, we have to remove the two rings Z 2 × Z 2 × Z 8 and Z 2 × Z 2 × Z 2 [x]/(x 3 ) from the list given in Theorem 4 [1]. Hence the revised statement of Theorem 4 [1] is as follows:…”

mentioning

confidence: 99%

See 3 more Smart Citations

Correction to: Classification of non-local rings with genus two zero-divisor graphs

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…There is no such vertex in Γ (Z 2 × Z 3 × Z 4 ) and hence wrongly included vertex (2, 0, 2) is deleted from the S 2 -embedding of Γ ( Z 2 × Z 3 × Z 4 ) given in Fig. 7 [1].…”

Section: Theorem 1 Let R Be a Non-local Commutative Ring Then The Genus Of The Zero-divisor Graph Is 2 If And Only If R Is Isomorphic To mentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

“…The purpose of this note is to correct some errors in the article [1]. The notation here will follow that of the original article.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Correction to: Classification of non-local rings with genus two zero-divisor graphs

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…Anderson and Livingston [2] modified Beck's definition and introduced the notion of zero-divisor graph. Surely, this is the most important graph associated with a ring, and not only zero-divisor graphs but also various generalizations of it have attracted many researchers; see for instance [1,7,13,8,5,4,10,16,17]. Therefore, this paper is devoted to introducing and studying a super graph of zero-divisor graphs.…”

Section: Introductionmentioning

confidence: 99%

The weakly zero-divisor graph of a commutative ring

Nikmehr¹,

Azadi²,

Nikandish³

2021

Revista De La Unión Matemática Argentina

View full text Add to dashboard Cite

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R is the undirected (simple) graph W Γ(R) with vertex set Z(R) * , and two distinct vertices x and y are adjacent if and only if there exist r ∈ ann(x) and s ∈ ann(y) such that rs = 0. It follows that W Γ(R) contains the zero-divisor graph Γ(R) as a subgraph. In this paper, the connectedness, diameter, and girth of W Γ(R) are investigated. Moreover, we determine all rings whose weakly zero-divisor graphs are star. We also give conditions under which weakly zero-divisor and zero-divisor graphs are identical. Finally, the chromatic number of W Γ(R) is studied.

show abstract