On realizing zero-divisor graphs of po-semirings

Source: arXiv

ABSTRACT 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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    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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May 29, 2014