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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 $$G$$.
##### MSC:
 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
##### Keywords:
monomial ideals; integral closure; bipartite graphs
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##### References:
 [1] TheNormalizTeam,Normaliz an open source tool for computations in a‰ne monoids etc.,available at: https://www.normaliz.uni-osnabrueck.de. [2] G.RestucciaandR.H.Villarreal,On the normality of monomial ideals of mixed products,Commun. Alg.29(2001), 3571-3580. · Zbl 1080.13509
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