Related papers: $F$-nilpotent rings and permanence properties

In this article we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen-Macaulay singularities. This is achieved by introducing…

We survey aspects of locally nilpotent linear groups. Then we obtain a new classification; namely, we classify the irreducible maximal locally nilpotent subgroups of $\mathrm{GL}(q, \mathbb F)$ for prime $q$ and any field $\mathbb F$.

We investigate the structure of locally finite profinite rings. We classify (Jacobson-) semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields, and we prove that the Jacobson…

In this article, a new and natural topology on the prime spectrum is established which behaves completely as the dual of the Zariski topology. It is called the flat topology. The basic and also some sophisticated properties of the flat…

We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show…

Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call…

We describe the support of $F$-finite $F$-modules over polynomial rings $R$ of prime characteristic. Our description yields an algorithm to compute the support of such modules; the complexity of our algorithm is also analyzed. To the best…

Let $R$ be a commutative Noetherian ring, $\Phi$ a system of ideals of $R$ and $I\in \Phi$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\dim M\leq 1$, then the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for all…

We define the concepts of weakly precious and precious rings which generalize the notions of weakly clean and nil-clean rings. We obtain some fundamental properties of these rings. We also obtain certain subclasses of such rings and then…

The aim of this paper is to introduce a new class of Noetherian rings of positive characteristic in terms of perfect closures and study their basic properties. If the perfect closure of a Noetherian ring is coherent, we call it an…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…

Smith and Van den Bergh introduced the notion of finite F-representation type as a characteristic $p$ analogue of the notion of finite representation type. In this paper, we prove two finiteness properties of rings with finite…

Let $D$ be a division ring with the center $F$ and $D^*$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^*$. We give some conditions that influence the existence of locally nilpotent…

We use our extension of the Noether-Lefschetz theorem to describe generators of the class groups at the local rings of singularities of very general hypersurfaces containing a fixed base locus. We give several applications, including (1)…

Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of…

We introduce the class weakly nil clean rings, as rings R in which for every a\in R there exist an idempotent e and a nilpotent q such that a-e-q\in eRa. Every weakly nil clean ring is exchange. Weakly nil clean rings contain pi-regular…

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper we give a short proof using filter regular sequences that the set of associated prime ideals of $H^t_I(R)$ is finite for any ideal $I$ and for any $t \ge 0$…

A new class of rings, {\em the class of weakly left localizable rings}, is introduced. A ring $R$ is called {\em weakly left localizable} if each non-nilpotent element of $R$ is invertible in some left localization $S^{-1}R$ of the ring…

Suppose that a Lie type algebra L over a field K admits a Frobenius group of automorphisms FH with cyclic kernel F of order n and complement H such that the fixed-point subalgebra of F is trivial and the fixed-point subalgebra of H is…

We investigate flat maps where the source or target is a Noetherian ring, giving necessary and/or sufficient conditions on a ring for such maps to exist. Along the way, we develop some general facts about flat ring maps, and exhibit many…