Related papers: A note on $UFD$

In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most…

In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also…

Let $A/K$ be an absolutely simple abelian surface defined over a number field $K$. We give unconditional upper bounds for the number of prime ideals $\mathfrak{p}$ of $K$ with norm up to $x$ such that $A$ has supersingular reduction at…

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 and positive depth. In this paper, we answer a question of…

In this paper, we prove that if $P$ is a homogeneous prime ideal inside a standard graded polynomial ring $S$ with $\dim(S/P)=d$, and for $s \leq d$, adjoining $s$ general linear forms to the prime ideal changes the $(d-s)$-th Hilbert…

We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise…

Let $(R,\fm)$ be a local ring and $\fa$ be an ideal of $R$. The inequalities $$\begin{array}{ll} \ \Ht(\fa) \leq \cd(\fa,R) \leq \ara(\fa) \leq l(\fa) \leq \mu(\fa) \end{array}$$ are known. It is an interesting and long-standing problem to…

We extend a result by Huneke and Watanabe bounding the multiplicity of $F$-pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of $F$-injective, generalized Cohen-Macaulay rings. We then…

The structure and the existence of maximal subrings in division rings are investigated. We see that if $R$ is a maximal subring of a division ring $D$ with center $F$ and $N(R)\neq U(R)\cup \{0\}$, where $N(R)$ is the normalizer of $R$ in…

For an odd prime p the cohomology ring of an elementary abelian p-group is polynomial tensor exterior. We show that the ideal of essential classes is the Steenrod closure of the class generating the top exterior power. As a module over the…

Inspired by a question raised by Eisenbud-Musta\c{t}\u{a}-Stillman regarding the injectivity of maps from ${\rm Ext}$ modules to local cohomology modules and the work by the third author with Pham, we introduce a class of rings which we…

Let $(\mathcal{O}_n, \mathfrak{m})$ denote the ring of germs of holomorphic functions $\mathbb{C}^n\to \mathbb{C}$, and let $I\subseteq \mathcal{O}_n$ be an $\mathfrak{m}$-primary ideal. Demailly and Pham showed that $\mathrm{lct}(I) \geq…

Let $R$ be a commutative $F$-algebra, where $F$ is a field of characteristic 0, satisfying the following conditions: $R$ is equidimensional of dimension $n$, every residual field with respect to a maximal ideal is an algebraic extension of…

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ and let $I=(f_1,...,f_s)$ be an ideal of $R.$ We prove that every associated prime $P$ of $H^i_I(R)$ satisfies $\text{dim}R/P\geqslant…

This paper is inspired by Michael Artin's paper "On The Join of Hensel Rings". In his paper, Artin proves that in an absolutely integrally closed ring the sum of two prime ideals is either prime or the whole ring. A more elementary proof of…

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the…

Fix an odd prime $p$. If $r$ is a positive integer and $f$ a polynomial with coefficients in $\mathbb{F}_{p^r}$, let $P_{p,r}(f)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_{p^r})$ that is periodic with respect to $f$. We show that as…

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…

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit…

The supremum of reduction numbers of ideals having principal reductions is expressed in terms of the integral degree, a new invariant of the ring, which is finite provided the ring has finite integral closure. As a consequence, one obtains…