By first describing the fixed maximal ideals of RL and those of its bounded part, R∗L, we show that every fixed maximal ideal of the bigger ring contracts to a fixed maximal ideal of the smaller ring, and every fixed maximal ideal of the smaller ring extends to a fixed maximal ideal of the bigger ring. However, the only instance where every maximal ideal of R∗L extends to a maximal ideal of RL ...

This result is a variation of a theorem of Duncan and O’Carroll [DO]: maximal ideals are replaced for any ideal using the unavoidable hypothesis dim(M/N) ≤ 1 (as an Example of Wang shows [W1]). Moreover it provides a partial positive answer to the question raised by Huneke in Conjecture 1.3 [H1]. We begin by recalling what is called uniform Artin-Rees properties. Let A be a noetherian ring, I b...

The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that J m = m. Vasconcelos conjectured that r(R/I) ≤ r(R/ in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture. 1. Reduction ...

In [1] a recursive warrant semantics for Defeasible Logic Programming extended with levels of possibilistic uncertainty for defeasible rules was introduced. The resulting argumentation framework, called RP-DeLP, is based on a general notion of collective (non-binary) conflict among arguments allowing to ensure direct and indirect consistency properties with respect to the strict knowledge. An o...

Let X be a compact set in the z-plane. We are interested in two function spaces associated with X: C(X) — space of all continuous complex-valued functions on X. P(X) =space of all uniform limits of polynomials on X. Thus a function ƒ on X lies in P{X) if there exists a sequence {Pn} of polynomials converging to ƒ uniformly on X. Clearly P(X) is part of C(X). QUESTION I. When is P(X) = C(X)t i.e...

It is known that the powers m of the maximal ideal of a local Noetherian ring share certain homological properties for all sufficiently large integers n. For example, the natural homomorphisms R → R/m are Golod, respectively, small, for all large n. We give effective bounds on the smallest integers n for which such properties begin to hold.

Let R denote a commutative local ring with maximal ideal m and residue field k — R/m. In this paper we determine the group automorphisms of the general linear group GL (R) when n > 3 and the characteristic of k is not 2.