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: Let R be a commutative ring with identity. The set I(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 I(R) is called the annihilating-ideal graph of R, denoted by AG(R). We write G for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in G 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 AG(R) ∈ G is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with AG(R) ∈ G . Also, a connection between finite rings and their corresponding graphs is realized.
    Communications in Algebra 03/2015; DOI:10.1080/00927872.2013.847950 · 0.36 Impact Factor
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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 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 II(R) is pure and shellable, where II(R) consists of all ideals of R.
    Frontiers of Mathematics in China 12/2014; 9(6). DOI:10.1007/s11464-014-0423-1 · 0.45 Impact Factor
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May 29, 2014