Ideals of reduction number two

Borel-fixed Ideals and Reduction Number

Deterministically Computing Reduction Numbers of Polynomial Ideals

We present two approaches to compute the (absolute) reduction number of a polynomial ideal. The first one puts the ideal into a position such that the reduction number of its leading ideal can be easily read off the minimal generators and then uses linear algebra to determine the reduction number of the ideal itself. The second method computes via a Gröbner system not only the absolute reductio...

Two Fσδ Ideals

We find two Fσδ ideals on N neither of which is Fσ whose quotient Boolean algebras are homogeneous but nonisomorphic. This solves a problem of Just and Krawczyk (1984). We consider Boolean algebras of the form P(N)/I, where I is an ideal on N containing the ideal Fin of finite sets. In [3] Just and Krawczyk formulated several conditions on the ideals I,J that guarantee their quotients P(N)/I an...

Borel Fixed Initial Ideals of Prime Ideals in Dimension Two

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger. Let R = k[x1, x2, . . . , xr] be a polynomial ring over a field. We will say that an ideal I ⊆ R has the saturated chain p...

Two Sided Ideals of Operators

1. Let X be a Banach space, and B(X) the Banach algebra of all bounded linear operators in X. The closed two sided ideals of B(X) (actually, of any Banach algebra) form a complete lattice L(X). Aside from very concrete cases, L(X) has not yet been determined; for instance, when X = l, l ^ p < « > , L(X) is a chain (i.e., totally ordered) with three elements: {o}, B(X) and the ideal C(X) of comp...