Related papers: Indivisible Sequences and Descendability

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their…

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with…

We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$.…

A new class of rings, the class of left localizable rings, is introduced. A ring $R$ is left localizable if each nonzero element of $R$ is invertible in some left localization $S^{-1}R$ of the ring $R$. Explicit criteria are given for a…

We consider trace ideals in Noetherian rings and focus our attention to one-dimensional analytically irreducible local rings. For such rings we classify those Gorenstein rings which admit only a finite number of trace ideals.

In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…

In this article, we have defined nil clean graph of a ring $R$. The vertex set is the ring $R$, two ring elements $a$ and $b$ are adjacent if and only if $a + b$ is nil clean in $R$. Graph theoretic properties like girth, dominating set,…

Given a set B of finite rooted graphs and a radius r as an input, we prove that it is undecidable to determine whether there exists a sequence (G_i) of finite bounded degree graphs such that the rooted r-radius neighbourhood of a random…

In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove…

We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an…

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is…

Let $p$ be a prime number. We define the notion of $F$-finiteness of homomorphisms of $\mathbb F_p$-algebras, and discuss some basic properties. In particular, we prove a sort of descent theorem on $F$-finiteness of homomorphisms of…

We discuss the projective line $\mathbb{P}(R)$ over a finite associative ring with unity. $\mathbb{P}(R)$ is naturally endowed with the symmetric and anti-reflexive relation "distant". We study the graph of this relation on $\mathbb{P}(R)$…

{Generalizing the notion of nil cleanness from \cite{D13}, in parallel to \cite{DM14}, we define the concept of {\it weak nil cleanness} for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition…

In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\,…

This paper aims at the following results: \begin{enumerate} \item The class of all $*$-regular rings forms a variety. \item A subdirectly irreducible $*$-regular ring $R$ is faithfully representable (i.e. isomorphic to a subring of an…

A path system $\mathcal{P}$ in a graph $G=(V,E)$ is said to be irreducible if there does not exist a partition $V= A\sqcup B$ such that $\mathcal{P}$ restricts to a path system on both $G[A]$ and $G[B]$. In this paper, we construct an…

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric…

We prove that, for every decreasing sequence {a \sb k} of natural numbers, there exists a map f: X --> X with cat (f\sp k)=a\sb k.

We gather some classical results and examples that show strict inclusion between the families of unital rings, rings with enough idempotents, rings with sets of local units, locally unital rings, s-unital rings and idempotent rings.