The adcending chain condition for real ideas.
The Apery algorithm for a plane singularity with two branches.
The Bordalo order on a commutative ring
If is a commutative ring with identity and is defined by letting mean or , then is a partially ordered ring. Necessary and sufficient conditions on are given for to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings of integers mod for . In particular, if is reduced, then is a lattice iff is a weak Baer ring, and is a distributive lattice iff is a Boolean ring, , , or a four element field.
The combinatorics of quiver representations
We give a description of faces, of all codimensions, for the cones spanned by the set of weights associated to the rings of semi-invariants of quivers. For a triple flag quiver and its faces of codimension 1 this description reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically...
The combinatorics of the Garsia-Haiman modules for hook shapes.
The cones associated to some tranversal polymatroids.
The connection between a conjecture of Carlisle and Kropholler, now a theorem of Benson and Crowley-Boevey, and Grothendieck's Riemannian-Roch and duality theorems.
The containment property for large quotient rings.
The covering of rings by valuation rings
The distributivity property of valuation rings
The Divisor Class Group of Ordinary and Symbolic Blow-ups.
The effect of quadratic transformations on degree functions.
The effect on associated prime ideals produced by an extension of the base field.
The equations of Rees algebras of ideals with linear presentation.
The finite free extension of artinian -algebras with the strong Lefschetz property
The First Isomorphism Theorem and Other Properties of Rings
Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial
The GCD property and irreducible quadratic polynomials.
The geometry of the set of characters iduced by valuations.
The group of automorphisms of a commutative algebra.
The group of divisibility of