Let I ⊂ K[x, y] be a monomial ideal. How small can µ(I 2 ) be in terms of µ(I)? It has been expected that the inequality µ(I 2 ) > µ(I) should hold whenever µ(I) ≥ 2. Here we disprove this expectation and provide a somewhat surprising answer to the above question.Theorem 1.1. For every integer m ≥ 5, there exists a monomial ideal I ⊂ K[x, y] such that µ(I) = m and µ(I 2 ) = 9.Moreover, this result is best possible for m ≥ 6.Theorem 1.2. Let I ⊂ K[x, y] be a monomial ideal. If µ(I) ≥ 6 then µ(I 2 ) ≥ 9.Here are some notation to be used throughout. We denote by M the set of monomials in K[x, y], i.e. M = {x i y j | i, j ∈ N}.As usual, we view M as partially ordered by divisibility. For a monomial ideal J ⊂ K[x, y], we denote by G(J) its unique minimal system of monomial generators. It is well known that G(J) is of cardinality µ(J) and consists of all monomials in J which are minimal under divisibility, i.e. G(J) = M ∩ J \ M ∩ J M * where M * = M \ {1}.
We study the number of generators of ideals in regular rings and ask the question whether µ(I) < µ(I 2 ) if I is not a principal ideal, where µ(J) denotes the number of generators of an ideal J. We provide lower bounds for the number of generators for the powers of an ideal and also show that the CM-type of I 2 is ≥ 3 if I is a monomial ideal of height n in K[x 1 , . . . , x n ] and n ≥ 3.2010 Mathematics Subject Classification. Primary 13C99; Secondary 13H05, 13H10.
In this paper we have some new results on sums of Hilbert space frames and Riesz bases. We also have a correction for some results in "S. Obeidat et al., Sums of Hilbert space frames,
We determine in an explicit way the depth of the fiber cone and its relation ideal for classes of monomial ideals in two variables. These classes include concave and convex ideals as well as symmetric ideals.2010 Mathematics Subject Classification. 05E40, 13A02, 13F20 .
Abstract. In this paper we introduce the concepts of atomic systems for operators and K -frames in Hilbert C * -modules and we establish some results.Mathematics subject classification (2010): 42C15, 46L05, 46H25.
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