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...

An ideal triangulation T of a hyperbolic 3-manifold M with one cusp is non-peripheral if no edge of T is homotopic to a curve in the boundary torus of M . For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of M . A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are id...

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

The regularity part follows from the primary decompositions part, so the heart of this paper is the analysis of the primary decompositions. In [S], this was proved for the primary components of height at most one over the ideal. This paper proves the existence of such a k but does not provide a formula for it. In the paper [SS], Karen E. Smith and myself find explicit k for ordinary and Frobeni...

a module m is called epi-retractable if every submodule of m is a homomorphic image of m. dually, a module m is called co-epi-retractable if it contains a copy of each of its factor modules. in special case, a ring r is called co-pli (resp. co-pri) if rr (resp. rr) is co-epi-retractable. it is proved that if r is a left principal right duo ring, then every left ideal of r is an epi-retractable ...

Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F ⊆ P (X) of size c, consisting of Borel sets which are not in I . Condition (M) states that there is a Borel function f : X → X with f[{x}] / ∈ ...

in this paper, various elementary properties of vague rings are obtained. furthermore, the concepts of vague subring, vague ideal, vague prime ideal and vague maximal ideal are introduced, and the validity of some relevant classical results in these settings are investigated.

Throughout we let (T,m, k) denote a commutative Noetherian local ring with maximal ideal m and residue field k. We let I ⊆ T be an ideal generated by a regular sequence of length c and set R := T/I. In the important paper [A], Avramov addresses the following question. Given a finitely generated R-module M , when does M have finite projective dimension over a ring of the form T/J , where J is ge...

Let R = ⊕ n≥0 Rn be a Noetherian homogeneous ring with onedimensional local base ring (R0,m0). Let q0 ⊆ R0 be an m0-primary ideal, let M be a finitely generated graded R-module and let i ∈ N0. Let Hi R+ (M) denote the i-th local cohomology module of M with respect to the irrelevant ideal R+ := ⊕ n>0 Rn of R. We show that the first Hilbert-Samuel coefficient e1 ( q0, Hi R+ (M)n ) of the n-th gra...