Related papers: Blowing Up Finitely Supported Complete Ideals in a…

We introduce and investigate a class of ring ideals, termed ring $\mathrm{M}$-ideals, inspired by the Alfsen--Effros theory of $\mathrm{M}$-ideals in Banach spaces. We show that $\mathrm{M}$-ideals extend the classical notion of essential…

In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{ \overline{I^rJ^s}\}$ of complete ${\mathfrak m}$-primary ideals and vanishing of the second normal…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

We investigate the containment problem of symbolic and ordinary powers of ideals in a commutative Noetherian domain $R$. Let $R$ be a normal domain of prime characteristic $p>0$ that is $F$-finite or essentially of finite type over an…

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type…

Let (R,m,k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R=d. Let M be a finitely generated R-module. We show that there exists a real number beta(M) such that lambda(M/I^[q]M) = e_{HK}(M) q^d…

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting Andr\'{e}'s perfectoid algebra techniques that, for complete local rings that have F-finite…

Let $A$ be a regular ring of dimension $d$ essentially of finite type over an infinite field $k$ of characteristic $\neq 2$. Let $P$ be a projective $A$-module of rank $n$ with $2n\geq d+3$. Let $I$ be an ideal of $A[T]$ of height $n$ and…

Let $T$ be a complete local ring. We present necessary and sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $A$ such that there exist height one prime ideals $\{J_k\}_{k = 1}^{\infty}$…

Let T be a complete local (Noetherian) equidimensional ring with maximal ideal m such that the Krull dimension of T is at least two and the depth of T is at least two. Suppose that no integer of T is a zerodivisor and that |T|=|T/m|. Let d…

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a…

In this paper, we show examples of local cohomology modules over ramified regular local ring, having finite set of associated primes. In doing so we consider our ramified regular local ring as Eisenstein extension of an unramified regular…

We prove an analogue of the Affine Horrocks' Theorem for local complete intersection ideals of height $n$ in $R[T]$, where $R$ is a regular domain of dimension $d$, which is essentially of finite type over an infinite perfect field of…

The Dedekind-Birkhoff theorem for finite-height modular lattices has previously been generalized to complete modular lattices using the theory of regular coverings. In this paper, we investigate regular coverings in lattices of filters and…

Let $(R,\mathfrak m)$ be an analytically unramified local ring of positive prime characteristic $p.$ For an ideal $I$, let $I^*$ denote its tight closure. We introduce the tight Hilbert function $H^*_I(n)=\ell(R/(I^n)^*)$ and the…

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of…

It is shown that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at some peak point. This answers a 31-year-old question of Joel Feinstein. Our example is R(K) for a certain compact…

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom_R(C,C) \cong R and Ext^i_R(C,C) = 0 for all i \geq 1. For certain local Cohen-Macaulay rings (R,m), we verify the equality of Hilbert-Samuel…

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the…

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal…