We deal here with the existence of half-way nonzero divisors for complete intersections and with related properties of their Hilbert functions.

Abstract. 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 Ngraded ring A of the form A ≥m := ℓ≥m A ℓ and monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of 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 of the form I(λ) := J(λ), where J(λ) = (x λ1

ABSTRACT. It is now possible to calculate the K"-theory of a large class of singular curves over fields of characteristic zero. Roughly speaking, the ÜT-theory of a curve is the X-theory of its (smooth) normalization plus a few shifted copies of the if-theory of the field plus a "nil part." The nil part is a vector space depending only on the analytic type of the singularities, and may be computed locally. We completely compute the nil part for seminormal curves and give a conjectural calculation in general which depends upon cyclic homology.Until recently, very little has been known about the higher algebraic Ktheory of anything but finite fields. In this note we announce the computability of the X-theory of singular curves in characteristic zero in terms of the K-theory of smooth curves and fields. If the curve is seminormal, we give a complete calculation; otherwise, the calculation depends on the validity of: CONJECTURE. Let B be a finite integral extension of a ring A, and let I be the conductor ideal. Assume A contains Q, the rational numbers. Then

It is natural to ask when a group has a planar Hasse lattice or more generally when its subgroup graph is planar. In this paper, we completely answer this question for finite groups. We analyze abelian groups, p-groups, solvable groups, and nonsolvable groups in turn. We find seven infinite families (four depending on two parameters, one on three, two on four), and three "sporadic" groups. In particular, we show that no nonabelian group whose order has three distinct prime factors can be planar.

No abstract