We generalize the strong Hanani-Tutte theorem to clustered graphs with two disjoint clusters, and show that an extension of our result to flat clustered graphs with three disjoint clusters is not possible. We also give a new and short proof for a result by Di Battista and Frati about efficient c-planarity testing of an embedded flat clustered graph with small faces based on the matroid intersec...

Introduction. The concept of a local ring was introduced by Krull [7](1), who defined such a ring as a commutative ring 9î in which every ideal has a finite basis and in which the set m of all non-units is an ideal, necessarily maximal. He proved that the intersection of all the powers of m is the zero ideal. If the powers of m are introduced as a system of neighborhoods of zero, then 3Î thus b...

A result of P. Samuel ([17] p. 186, Chap.II, Théorème 5) says that in a local noetherian ring (O,M) of Krull dimension d in which the residual field k is infinite, the multiplicity of a M-primary ideal I is equal to the multiplicity of an ideal (x1, . . . , xd) generated by some parameter sequence x1, . . . , xd contained in I. By a theorem of Rees ([16] p.142 Theorem 9.44), this implies that t...

In this paper we study wide subcategories. A full subcategory of R-modules is said to be wide if it is abelian and closed under extensions. Hovey [Hov01] gave a classification of wide subcategories of finitely presented modules over regular coherent rings in terms of certain specialisation closed subsets of Spec(R). We use this classification theorem to study K-theory and “Krull-Schmidt” decomp...

Answering a long-standing question in the theory of torsion modules, we show that weakly productively bounded domains are necessarily productively bounded. (See the introduction for definitions.) Moreover, we prove a twin result for the ideal lattice L of a domain equating weak and strong global intersection conditions for families (Xi)i∈I of subsets of L with the property that i∈I Ai = 0 whene...

Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N∗(x) of an element x ∈ N to be bounded by the rank of N . The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x ∈ mN . We show that the inequality even holds if the hypothesis becomes true after first extending sc...