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Integral closure of bipartite graph ideals. (English) Zbl 1445.13012
Following I. Swanson and C. Huneke [Integral closure of ideals, rings, and modules. Cambridge: Cambridge University Press (2006; Zbl 1117.13001)], for an ideal \(I\) of a commutative ring \(R\), the integral closure \(\overline{I}\) of \(I\) consists of all elements \(r\) in \(R\) that are integral over \(I\), i.e., for such an element \(r\), there exist an integer \(n\) and elements \(a_i\in I^i\) (\(1\leq i\leq n\)) such that \(r^n+\sum_{i=1}^na_ir^{n-i}=0\) holds. Recall that \(\overline{I}\) is an ideal and, it is monomial if \(I\) is a monomial ideal of a polynomial ring \(\mathbb F[X]\). For a monomial ideal \(I\), \(\overline{I}\) is a monomial ideal generated by all monomials \(u\) for which there exists an integer \(k\) such that \(u^k\in I^k\).
In the paper under review, the authors introduce a log set \(log(F)\) for any set \(F\) of monomials, and use it to describe the integral closure of an edge ideal of a complete bipartite graph. The authors also studied the related properties of generalized graph ideals of a strong complete quasi-bipartite graph \(G\), which has a loop in each vertex, in addition to the complete bipartite structure on \(G\).
13B22 Integral closure of commutative rings and ideals
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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[1] TheNormalizTeam,Normaliz an open source tool for computations in a‰ne monoids etc.,available at: https://www.normaliz.uni-osnabrueck.de.
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