On realizing zero-divisor graphs of po-semirings

Source: arXiv


In this paper, we determine bipartite graphs and complete graphs with horns,
which are realizable as zero-divisor graphs of po-semirings. As applications,
we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb
{AG}(R)$ are either bipartite graphs or complete graphs with horns.

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    • "(3) It is natural to ask the following question: Can any two-star graph be realized as the zero divisor graph of a po-semiring when condition (C 3 ) is dropped? The answer is yes and see [14] for a complete answer. "
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    ABSTRACT: A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A po-semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a po-semiring are studied. In particular, it is proved that under some mild assumption the set $Z(A)$ of nonzero zero divisors of $A$ is $A\setminus \{0,1\}$, each prime element of $A$ is a maximal element, and the zero divisor graph $\G(A)$ of $A$ is a finite graph if and only if $A$ is finite. For a po-semiring $A$ with $Z(A)=A\setminus \{0,1\}$, 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 applications of prime elements, it is shown that the structure of a po-semiring $A$ is completely determined by the structure of integral po-semirings if either $|Z(A)|=1$ or $|Z(A)|=2$ and $Z(A)^2\not=0$. Applications to the ideal structure of commutative rings are considered. This work will appear in Front. Math. China (added in Aug.1, 2014)
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    ABSTRACT: In this paper, we study some graphs which are realizable and some which are not realizable as the incomparability graph (denoted by Γ′(L)Γ′(L)) of a lattice L with at least two atoms. We prove that the complete graph KnKn with two horns is realizable as Γ′(L)Γ′(L). We show that the complete graph K3K3 with three horns is not realizable as Γ′(L)Γ′(L), however it is realizable as the zero-divisor graph of L. Also we give a necessary and sufficient condition for a complete bipartite graph with one horn and with two horns to be realizable as Γ′(L)Γ′(L) for some lattice L.
    Journal of Discrete Algorithms 11/2013; 23:63–75. DOI:10.1016/j.jda.2013.07.001
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