This paper is a simple summary of the first most basic definitions in Algebraic Geometry as they are presented in Dummit and Foote ([1]), with a focus on establishing a dictionary between algebra and geometry. In particular this dictionary helps in visualizing certain computations that take place in polynomial rings. Among these are decomposing ideals in a manner analogous to prime factorizatio...

We survey and extend recent work on integrally closed overrings of two-dimensional Noetherian domains, where such overrings are viewed as intersections of valuation overrings. Of particular interest are the cases where the domain can be represented uniquely by an irredundant intersection of valuation rings, and when the valuation rings can be chosen from a Noetherian subspace of the Zariski-Rie...

Let R be a local, Noetherian ring and I ⊆ R an ideal. A question of Kodiyalam asks whether for fixed i > 0, the polynomial giving the ith Betti number of In has degree equal to the analytic spread of I minus one. Under mild conditions on R, we show that the answer is positive in a number of cases, including when I is divisible by m or I is an integrally closed m-primary ideal.

The models of normal open induction are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano’s induction axioms for open formulas in the language of ordered semirings. It is known that neither open induction nor the usually studied stronger fragments of arithmetic (where induction for quantified formulas is allowed), have the joint embed...

Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano’s axioms for open formulas in the language of ordered semirings. (Where normal means integrally closed in its fraction field.) In 1964 Shepherdson gave a recursive nonstandard model of open induction. His model is not normal and does not have any infinite prime elements. In this paper ...

Let (R,m) be a Noetherian local ring and let I ⊆ R be an ideal. This paper studies the question of when mI is integrally closed. Particular attention is focused on the case R is a regular local ring and I is a reduced ideal. This question arose through a question posed by Eisenbud and Mazur on the existence of evolutions. Let (R,m) be a Noetherian local ring and let I ⊆ R be an ideal. In this p...

This paper seeks ring-theoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S...

Suppose D is an integral domain with quotient eld K and that L is an extension eld of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by All rings considered in this paper are assumed to...

Abstract Let $(A,\mathfrak m)$ be an excellent two-dimensional normal local domain. In this paper, we study the elliptic and strongly ideals of A with aim to characterize singularities, according definitions given by Wagreich Yau. analogy rational in main result, a singularity terms Hilbert coefficients integrally closed $\mathfrak m$ -primary . Unlike $p_g$ -ideals, are not necessarily necessa...