Dimensional Local Ring Research Articles

Constructive General Neron Desingularization for one dimensional local rings

An algorithmic proof of General Neron Desingularization is given here for one dimensional local rings and it is implemented in Singular. Also a theorem recalling Greenberg' strong approximation theorem is presented for one dimensional local rings.

One dimensional local rings of maximal and almost maximal length

Let ( A, m, k) denote a one dimensional, Cohen-Macaulay, local ring with maximal ideal m and residue class field k. We assume A is a reduced, excellent local ring which has a canonical module ω A . With no loss of generality, we can also assume k is infinite. Let Ā denote the integral closure of A in its total quotient ring Q, and let b denote the conductor of A in Ā. A has maximal length if l A( A ̄ A ) = l A( A b ) t(A) Here t( A) denotes the Cohen-Macaulay type of A, and l A(∗) denotes the length of the A-module ∗. A has almost maximal length if l A( A ̄ A ) = l A( A b ) t(A) − 1 The two main results of this paper are as follows: A has maximal length if and only if either A is Gorenstein or there exists an x ϵ m and a positive integer p such that m = (x, b), andb A ̄ = x p A ̄ . Let e( A) denote the multiplicity of A, Then A has almost maximal length, and 1 + t( A) = e( A) if and only if there exists a transversal x of m such that m = (x, b), and l A( b x p A ̄ ) = 1 . Here p = min{i¦x iϵb} These two theorems generalize more specific results for semigroup rings obtained in Brown and Curtis (“Numerical Semigroups of Maximal and Almost Maximal Length,” Semigroup Forum, Vol. 42, Springer-Verlag, Berlin/New York, 1991, 219–235).

On Berger's conjecture about one dimensional local rings

On Berger's conjecture about one dimensional local rings

Arithmetic on two dimensional local rings

Arithmetic on two dimensional local rings

On the finite generation of symbolic blow-ups

"symbolic blow-up", (~ P(n)=S a Noetherian ring? If S is Noetherian and dim n 0 RIP=l, Cowsik observed that P must be a set-theoretic complete intersection. If R is not regular, there is no hope in general of S being Noetherian. In fact if R is any normal two dimensional local ring and P is a height one prime ideal of R which is not torsion in the class group of R, then S is never Noetherian. This idea is behind the Rees counterexample [6] to the generalized Zariski question. As far as the author knows, there are no known counter examples to the question of Cowsik. The most trivial case is the following: if R is Nagata and

Boundedness in Two Dimensional Local Rings

Boundedness in Two Dimensional Local Rings

Semifactoriality and Muhly's condition ( N) in two dimensional local rings

Semifactoriality and Muhly's condition ( N) in two dimensional local rings

Generalized primary rings

The purpose of this paper is to investigate the properties of commutative rings with identity in which all ideals are primary. We call such rings generalized primary rings. Zariski primary rings [8; p. 204], Snapper's completely primary rings [7] and discrete valuation domains are generalized primary. It is shown in the example constructed in 2.4 that generalized primary rings need not be primary rings. There are also examples of generalized primary rings which are not discrete valuation rings (4.11). The principal results concerning generalized primary rings are summarized here. Generalized primary rings are local rings (Theorem 2.3) and a Noetherian domain is generalized primary iff it is a one dimensional local ring (Theorem 4.5). If the requirement that the ring is a domain is dropped, then this result need not be true (4.7). A commutative hereditary ring with identity is generalized primary iff it is a discrete valuation domain (Theorem 4.9). A Noetherian generalized primary is a discrete valuation ring iff the unique maximal ideal is principal (Theorem 4.11). The importance of the first two results mentioned above can be realized by noting that the ring of integers is one dimensional Noetherian hereditary domain but is not a generalized primary ring. Most of the results in this paper are obtained by using the concept of the radical. For this we refer the reader to [8; p. 147]. V ~ represents the radical of an ideal A and ] /~ is the intersection of all prime ideals containing A E2; p. 253].