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- William J. Heinzer, LOUIS J. RATLIFF
- 2010

Let I be a proper nonnilpotent ideal in a local (Noetherian) ring (R,M) and let J be a reduction of I; that is, J ⊆ I and JIn = In+1 for some nonnegative integer n. We prove that there exists a finite free local unramified extension ring S of R such that the ideal IS has a minimal reduction K ⊆ JS with the property that the number of elements in a minimal… (More)

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B ⊆ K of A that are wellcentered on A in the sense that each principal ideal of B is generated by an element of A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A. If B = A[b] is a… (More)

For a regular ideal having a principal reduction in a Noetherian ring we consider the structural numbers that arise from taking the Ratliff-Rush closure of the ideal and its powers. In particular, we analyze the interconnections among these numbers and the relation type and reduction number of the ideal. We prove that certain inequalites hold in general… (More)

- William J. Heinzer, Bernard Johnston, David Lantz

Let I be an m-primary ideal of a Noetherian local ring (R,m). We consider the Gorenstein and complete intersection properties of the associated graded ring G(I) and the fiber cone F (I) of I as reflected in their defining ideals as homomorphic images of polynomial rings over R/I and R/m respectively. In case all the higher conormal modules of I are free… (More)

- William J. Heinzer, Christel Rotthaus, Sylvia M. Wiegand

C ↪→ D1 := k[x] [[y/x]] ↪→ · · · ↪→ Dn := k[x] [[y/x]] ↪→ · · · ↪→ E. (2) With regard to Equation 2, for n a positive integer, the map C ↪→ Dn is not flat, but Dn ↪→ E is a localization followed by an adic completion of a Noetherian ring and therefore is flat. We discuss the spectra of these rings and consider the maps induced on the spectra by the… (More)

- Marco D’Anna, Anna Guerrieri, William J. Heinzer, Jim Huckaba

For a regular ideal I having a principal reduction in a Noetherian local ring (R,m) we consider properties of the powers of I as reflected in the fiber cone F (I) and the associated graded ring G(I) of I. In particular, we examine the postulation number of F (I) and compare it with the reduction number of I, and the postulation number of G(I) when the… (More)

We study content ideals of polynomials and their behavior under multiplication. We give a generalization of the Lemma of Dedekind–Mertens and prove the converse under suitable dimensionality restrictions.

- William J. Heinzer, Bernard Johnston, David Lantz

For an ideal I of a Noetherian local ring (R,m) we consider properties of I and its powers as reflected in the fiber cone F (I) of I . In particular, we examine behavior of the fiber cone under homomorphic image R→ R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψJ : FR(I) → FR′(IR). We consider the… (More)