相关论文: Tilting and untilting for ideals in perfectoid rin…

Let $R$ be a commutative ring, $Y\subseteq \mathrm{Spec}(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in…

It is proved that the ring $R$ with center $Z(R)$, such that the module $R_{Z(R)}$ is an essential extension of the module $Z(R)_{Z(R)}$, is not necessarily right quasi-invariant, i.e., maximal right ideals of the ring $R$ are not…

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic…

Let $R$ be a commutative ring with unity. The prime ideal sum graph $\text{PIS}(R)$ of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are…

Let $I$ be a regular proper ideal in a Noetherian ring $R$, let $e \ge 2$ be an integer, let $\mathbf T_e = R[u,tI,u^{\frac{1}{e}}]' \cap R[u^{\frac{1}{e}},t^{\frac{1}{e}}]$ (where $t$ is an indeterminate and $u =\frac{1}{t}$), and let…

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with the vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ and…

An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive…

For a commutative ring $R$ with unit $1\ne 0$ and a multiplicatively closed subset $S$ of $R$, we introduce a new topology on the $S$-prime spectrum $\mathrm{Spec}_SR$ of $R$ called the $S$-flat topology. Our aims is to give an algebraic…

We construct a map from the prestack of Tate objects over a commutative ring $k$ to the stack of $\mathbb{G}_{\rm m}$-gerbes. The result is obtained by combining the determinant map from the stack of perfect complexes as proposed by…

In this second paper, we study the case of substitution tilings of R^d. The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j=0, ..., d-1. We reconstruct the tiling's equivalence relation in a purely…

Let $R$ be a commutative ring with identity and $\Bbb A (R)$ be the set of ideals of $R$ with non-zero annihilator. The annihilator-ideal graph of $R$, denoted by $A_{I} (R) $, is a simple graph with the vertex set $\Bbb A(R)^{\ast} := \Bbb…

\noindent The algebraic characterization of classes of locally isomorphic aperiodic tilings, being examples of quantum spaces, is conducted for a certain type of tilings in a manner proposed by A. Connes. These $2$-dimensional tilings are…

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., $\{0\})$ that are closed under the natural metric, but has no prime ideals closed under that metric; hence closed…

Let X be a projective variety, $\sigma$ an automorphism of X, L a $\sigma$-ample invertible sheaf on X, and Z a closed subscheme of X. Inside the twisted homogeneous coordinate ring $B = B(X, L, \sigma)$, let I be the right ideal of…

We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically…

Let $\Omega \subseteq {\bf R}^d$ be an open set of measure 1. An open set $D \subseteq {\bf R}^d$ is called a ``tight orthogonal packing region'' for $\Omega$ if $D-D$ does not intersect the zeros of the Fourier Transform of the indicator…

We consider the lattice-ordered groups Inv$(R)$ and Div$(R)$ of invertible and divisorial fractional ideals of a completely integrally closed Pr\"ufer domain. We prove that Div$(R)$ is the completion of the group Inv$(R)$, and we show there…

Let $\mathcal{I},\mathcal{J}$ be two ideals on $\mathbf{N}$ which contain the family $\mathrm{Fin}$ of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix $A=(a_{n,k})$ which maps…

In this paper, we use trace methods to study the algebraic $K$-theory of rings of the form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. We compute the relative $p$-adic $K$ groups for $R$ a perfectoid ring. In particular, we get the integral…

Let R be an excellent local ring, m its maximal ideal and I an ideal. Then there exists a positive integer c such that for all integers n, the integral closure of (I + m^n) is contained in m^(n/c) + the integral closure of I. In the proof,…