On $n$-trivial extensions of rings

Anderson, D. D.; Bennis, Driss; Fahid, Brahim; Shaiea, Abdulaziz

doi:10.1216/rmj-2017-47-8-2439

Cited by 16 publications

(19 citation statements)

References 24 publications

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“…Now we are in a position to give the desired example. For that we use a new ring construction recently introduced in [1].…”

Section: 2)mentioning

confidence: 99%

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On the extended zero divisor graph of commutative rings

Bennis¹,

Mikram²,

Taraza³

2016

Turk J Math

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In this paper we present a new graph that is closely related to the classical zero-divisor graph. In our case two nonzero distinct zero divisors x and y of a commutative ring R are adjacent whenever there exist two nonnegative integers n and m such that x n y m = 0 with x n ̸ = 0 and y m ̸ = 0. This yields an extension of the classical zero divisor graph Γ(R) of R , which will be denoted by Γ(R). First we distinguish when Γ(R) and Γ(R) coincide. Various examples in this context are given. We show that if Γ(R) ̸ = Γ(R) , then Γ(R) must contain a cycle. We also show that if Γ(R) ̸ = Γ(R) and Γ(R) is complemented, then the total quotient ring of R is zero-dimensional. Among other things, the diameter and girth of Γ(R) are also studied.

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“…Now we are in a position to give the desired example. For that we use a new ring construction recently introduced in [1].…”

Section: 2)mentioning

confidence: 99%

“…Then x ∈ Ann(y n ) = Ann(y) by Lemma 2.2 and therefore xy = 0 . (3) ⇒(1). Let x and y be two adjacent vertices in Γ(R).…”

mentioning

confidence: 99%

On the extended zero divisor graph of commutative rings

Bennis¹,

Mikram²,

Taraza³

2016

Turk J Math

Self Cite

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show abstract

“…The ring 1 in the proof of Theorem 6(2) is isomorphic to the 2-trivial extension Z⋉ 2 F ⋉F (in the sense of [26]). The theory developed in [26] can illuminate such constructions.…”

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confidence: 99%

“…The ring 1 in the proof of Theorem 6(2) is isomorphic to the 2-trivial extension Z⋉ 2 F ⋉F (in the sense of [26]). The theory developed in [26] can illuminate such constructions. For instance, the assertion that 1 is a prime ideal of 1 in the above-mentioned proof can be seen via [26,Theorem 4.7].…”

mentioning

confidence: 99%