Minimal prime ideals and cycles in annihilating-ideal graphs

Aalipour, Ghodratollah; Akbari, S.; Nikandish, Reza; Nikmehr, M. J.; Shaveisi, Farzad

doi:10.1216/rmj-2013-43-5-1415

Cited by 26 publications

(18 citation statements)

References 12 publications

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“…By Corollary 3.4, R is a local ring with exactly two nontrivial ideals. Since J(R) 2 = J(R) by the Nakayama Lemma, J(R) and J(R) 2 are the nontrivial ideals of R by Theorem 3.3 (2). Hence, J(R) 3 = 0 and J(R) 2 = 0.…”

Section: Corollary 35mentioning

confidence: 89%

“…(1) If c and u are incomparable, then c + u ∈ A * 1 and Z(A) 2 = 0 by Lemma 3.1 (2). Assume c + u = a 0 .…”

Section: Structure Of a Bounded Semiring A With Small |Z(a)|mentioning

confidence: 98%

“…The other statements also follow directly from Lemma 3.1 (2). (2) In what follows, we assume that c and u are comparable with c < u. Then c 2 = 0, and c is the least nonzero element of A.…”

Section: Structure Of a Bounded Semiring A With Small |Z(a)|mentioning

confidence: 99%

“…Applying Corollary 3.4 to the bounded semiring I(R), we obtain the following corollary, Corollary 3.5, which is essentially the same as [1, Theorem 3] (by [12,Theorem 2.1]). For the definition and some known results on the annihilating ideal graph AG(R), the reader is referred to [1,2,5]. Note that the graph AG(R) defined in [5] is exactly the zero-divisor graph of the multiplicative semigroup I(R).…”

Section: Structure Of a Bounded Semiring A With Small |Z(a)|mentioning

confidence: 99%

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Zero divisors and prime elements of bounded semirings

Li²,

Lu³

2014

Front. Math. China

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A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z(A) of nonzero zero divisors of A is A \ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z(A) = A \ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either |Z(A)| = 1 or |Z(A)| = 2 and Z(A) 2 = 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I(R) is pure and shellable, where I(R) consists of all ideals of R.

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Section: Corollary 35mentioning

confidence: 89%

“…(1) If c and u are incomparable, then c + u ∈ A * 1 and Z(A) 2 = 0 by Lemma 3.1 (2). Assume c + u = a 0 .…”

Section: Structure Of a Bounded Semiring A With Small |Z(a)|mentioning

confidence: 98%

“…The other statements also follow directly from Lemma 3.1 (2). (2) In what follows, we assume that c and u are comparable with c < u. Then c 2 = 0, and c is the least nonzero element of A.…”

Section: Structure Of a Bounded Semiring A With Small |Z(a)|mentioning

confidence: 99%

Section: Structure Of a Bounded Semiring A With Small |Z(a)|mentioning

confidence: 99%

See 2 more Smart Citations

Zero divisors and prime elements of bounded semirings

Li²,

Lu³

2014

Front. Math. China

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“…There are many papers on assigning a graph to a ring R, see [1][2][3][4]9,10]. In this paper, we introduce the M-principal graph of R, denoted by M − PG(R), where M is an R-module.…”

Section: Introductionmentioning

confidence: 99%

The M-principal graph of a commutative ring

Nikmehr

Heydari

2014

Period Math Hung

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Let R be a commutative ring and M be an R-module. In this paper, we introduce the M-principal graph of R, denoted by M − PG(R). It is the graph whose vertex set is R\{0}, and two distinct vertices x and y are adjacent if and only if x M = y M. In the special case that M = R, M − PG(R) is denoted by PG(R). The basic properties and possible structures of these two graphs are studied. Also, some relations between PG(R) and M − PG(R) are established.

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Strong metric dimension in annihilating-ideal graph of commutative rings

Jalali

Nikandish

2022

AAECC

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Minimal prime ideals and cycles in annihilating-ideal graphs

Cited by 26 publications

References 12 publications

Zero divisors and prime elements of bounded semirings

Zero divisors and prime elements of bounded semirings

The M-principal graph of a commutative ring

Strong metric dimension in annihilating-ideal graph of commutative rings

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