Integral closure of bipartite graph ideals. (English) Zbl 1445.13012

Imbesi, Maurizio; La Barbiera, Monica

Kodai Math. J. 43, No. 2, 243-255 (2020).

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 \(G\).

Reviewer: Tongsuo Wu (Shanghai)

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

Software:

Normaliz

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