For a local ring $(R,\M) $ of infinite residue field and positive depth, we address the question
raised by H. Dao on how to control the asymptotic behaviour of the $\M $-full, full, and …

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full
properties of certain product ideals in a Noetherian local ring $ R $ with infinite residue field …

In the present paper, we study the Dao numbers $\mathfrak {d} _1 (I),\mathfrak {d} _2 (I) $
and $\mathfrak {d} _3 (I) $ of an ideal $ I $ of a Noetherian local ring $(R,\mathfrak {m}, K) …

Let $(R,\m, k) $ be a local (Noetherian) ring of positive prime characteristic $ p $ and
dimension $ d $. Let $ G_\dt $ be a minimal resolution of the residue field $ k $, and for each …

We give many new results related to the theory of tight closure and its generalizations.
Explicitly, we establish a series of results showing that the Jacobian ideal is contained in the …

If R is a polynomial ring over a field k, M is a finitely generated graded R-module (for the
standard grading of R) and m:= R> 0, the invariants ai (M):= inf {µ| Hi m (M)> µ= 0} and bi …

We prove that limits of multiplicities associated to graded families of ideals exist under very
general conditions. Most of our results hold for analytically unramified equicharacteristic …

We expand the notion of core to cl-core for Nakayama closures cl. In the characteristic p> 0
setting, when cl is the tight closure, denoted by*, we give some examples of ideals when the …

Consider a Cohen-Macaulay local ring $(R,\mathfrak m) $ with dimension $ d\geq 2$, and
let $ I\subseteq R $ be an $\mathfrak m $-primary ideal. Denote $ r_ {J}(I) $ as the reduction …

Let R be a Noetherian ring of positive characteristic p. For every ideal a in R, and for every
ideal J whose radical contains a, one can define asymptotic invariants that measure the …