Related papers: Structural results on harmonic rings and lessened …

In this paper, we formulate and prove a general compactness theorem for harmonic maps using Deligne-Mumford moduli space and families of curves. The main theorem shows that given a sequence of harmonic maps over a sequence of complex…

Let $D$ be an integrally closed domain with quotient field $K$. Let $A$ be a torsion-free $D$-algebra that is finitely generated as a $D$-module. For every $a$ in $A$ we consider its minimal polynomial $\mu_a(X)\in D[X]$, i.e. the monic…

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical…

A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute. By {\rm SRC} factorization, Borooah, Diesl, and Dorsey \cite{BDD051} completely determined when ${\mathbb M}_n(R)$ over a…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be a uniformly $S$-Artinian ($u$-$S$-Artinian for abbreviation) module if there is $s\in S$ such that any descending chain of…

Let $k$ be a division ring and let $G$ be either a torsion-free virtually compact special group or a finitely generated torsion-free $3$-manifold group. We embed the group algebra $kG$ in a division ring and prove that the embedding is…

We consider and study those rings in which each nil-clean or clean element is uniquely nil-clean. We establish that, for abelian rings, these rings have a satisfactory description and even it is shown that the classes of abelian rings and…

We study clean group rings and also the group rings whose every element is a sum of two units. We also prove that if R is an Abelian exchange ring and G is a locally finite group, then the group ring RG has stable range one.

Several results in functional analysis are extended to the setting of $L^0$-modules, where $L^0$ denotes the ring of all measurable functions $x\colon \Omega\to \mathbb{R}$. The focus is on results involving compactness. To this end, a…

Some variations of $\pi$-regular and nil clean rings were recently introduced in \cite{5,8,7}, respectively. In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that $(m,…

In this article, we first generalize Kaplansky's zero-divisor conjecture of group-rings $K[G]$ (with $K$ a field) to the more general setting of $G$-graded rings $R=\bigoplus\limits_{n\in G}R_{n}$ with $G$ a torsion-free group. Then we…

A ring is called $n$-perfect ($n\geq 0$), if every flat module has projective dimension less or equal than $n$. In this paper, we show that the $n$-perfectness relate, via homological approach, some homological dimension of rings. We study…

We introduce the notion of a projectively simple ring, which is an infinite-dimensional graded k-algebra A such that every 2-sided ideal has finite codimension in A (over the base field k). Under some (relatively mild) additional…

Let $(M^n, g)(n\geq3)$ be an $n$-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by $R$ and $\mathring{Rm}$ the scalar curvature and the trace-free Riemannian curvature tensor of $M$,…

We consider the KZ connection associated with a family of hyperelliptic curves of genus $g$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then the dual connection is the Gauss-Manin connection of that family. We observe that the…

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require its surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure…

A ring $R$ is nil-clean if every element in $R$ is the sum of an idempotent and a nilpotent. A ring $R$ is abelian if every idempotent is central. We prove that if $R$ is abelian then $M_n(R)$ is nil-clean if and only if $R/J(R)$ is Boolean…

An example is given of a UFD which has infinitely generated Derksen invariant. The ring is \textquotedblleft almost rigid\textquotedblright\ meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that…

The main result here is that a simple separable C*-algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the…

In the preprint arXiv:2511.07900 we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative…