The Classification of the Annihilating-Ideal Graphs of Commutative Rings

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

doi:10.1142/s1005386714000200

Cited by 40 publications

(29 citation statements)

References 9 publications

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“…Then p is not nilpotent. By Theorem 2.2, there exists a positive integer r such that p r is annihilated by a nonzero idempotent, say, f. The rest of the proof is almost the same as in (1). Note that p r (g 1 g 2 · · · g s ) q implies that p r (g 1 g 2 · · · g n ) is not nilpotent.…”

Section: Theorem 27 For a Bounded Semiring A Each Prime Element Of mentioning

confidence: 84%

“…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: Theorem 27 For a Bounded Semiring A Each Prime Element Of mentioning

confidence: 84%

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

confidence: 99%

Zero divisors and prime elements of bounded semirings

Li²,

Lu³

2014

Front. Math. China

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“…In recent years, assigning graphs to rings has played an important role in the study of structures of rings, see for instance [1,2,[4][5][6]. The benefit of studying these graphs is that one may find some results about the algebraic structures and the vice versa.…”

Section: Introductionmentioning

confidence: 99%

Commutative Rings R Whose C(𝔸𝔾(R)) Consist of Only Triangles

2015

Communications in Algebra

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Let R be a commutative ring with identity. The set R of all ideals of R is a bounded semiring with respect to ordinary addition, multiplication and inclusion of ideals. The zero-divisor graph of R is called the annihilating-ideal graph of R, denoted by R . We write for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in that can be realized as either the zero-divisor graphs of bounded semirings or the annihilating-ideal graphs of commutative rings are determined. A necessary and sufficient condition for a ring R such that R ∈ is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with R ∈ . Also, a connection between finite rings and their corresponding graphs is realized.

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“…Some authors have also extended the graph of zero-divisors to non-commutative rings, see [18] and [2]. In [1], [12], and [13], the graph of zero-divisors for commutative rings has been generalized to the annihilating-ideal graph of commutative rings (two ideals I and J are adjacent if IJ = (0)). In [11], the classic zero-divisor graph has been generalized to modules over commutative rings.…”

Section: Introductionmentioning

confidence: 99%