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In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of $h$-wheel graphs, under certain condition, are normal.

In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals., Comment: 15 pages, 2 figures; pre-published version

In this paper, our main focus is to explore different classes of nearly normally torsion-free ideals. We first characterize all finite simple connected graphs with nearly normally torsion-free cover ideals. Next, we characterize all normally torsion-free $t$-spread principal Borel ideals that can also be viewed as edge ideals of uniform multipartite hypergraphs.

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by all homogeneous elements of degree at least m and monomial ideals in a polynomial ring over a field. For ideals of the first trype we generalize a recent result of S. Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates over a field is normal if and only if the first n-1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals obtained by taking integral closures of m-primary ideals generated by powers of the variables. We obtain classes of normal monomial ideals and arithmetic critera for deciding when the monomial ideal is not normal., Comment: 19 pages