Related papers: Uniqueness of real closure * of Baer regular rings
In this paper we give a characterisation of real closure * of regular rings, which is quite similar to the characterisation of real closure * of Baer regular rings seen in [4]. We also characterize Baer-ness of regular rings using near-open…
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…
Here we try to distinguish and compare different notions of real closedness mainly one developed by N. Schwartz in his Habilitationschrift and the other developed by A. Sankaranand K. Varadarajan which we shall call real closed *. We stick…
Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative…
We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that…
Absolute integral closures of general commutative unital rings are explored. All rings admit absolute integral closures, but in general they are not unique. Among the reduced rings with finitely many minimal prime ideals, finite products of…
We prove that a complete local or graded one-dimensional domain of prime characteristic has finite F-representation type if its residue field is algebraically closed or finite, and present examples of a complete local or graded…
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…
In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of…
We consider a class ${\mathcal C}$ of Baer *-rings (also treated in [S. K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag, Berlin-Heidelberg-New York, 1972.] and [L. Va\v{s}, Dimension and…
We characterize the rings in which the equality $(\tau I:\tau)= I^*$ holds for every ideal $I \subset R$. Under certain assumptions, these rings must be either weakly F-regular or one-dimensional.
A ring is clean (resp. almost clean) if each of its elements is the sum of a unit (resp. regular element) and an idempotent. In this paper we define the analogous notion for *-rings: a *-ring is *-clean (resp. almost *-clean) if its every…
The notion of strict closedness of rings was given by J. Lipman in connection with a conjecture of O. Zariski. The present purpose is to give a practical method of construction of strictly closed rings. It is also shown that the…
We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.
An $R$-algebra $S$ is $R$-solid if there exists a nonzero $R$-linear map $S \rightarrow R$. In characteristic $p$, the study of $F$-singularities such as Frobenius splittings implicitly rely on the $R$-solidity of $R^{1/p}$. Following…
This paper mainly focuses on commutative local domains of dimension one. We then obtain a criterion for a ring to have a finite number of trace ideals in terms of integrally closed ideals. We also explore properties of such rings related to…
The set of minimal primes of a ring is a very important set as far the spectrum of a ring is concerned as every prime contains a minimal prime. So, knowing the minimal primes is the first (important and difficult) step in describing the…
This article has the following aims: (1) Extend the notion of fuchsian singularities (of first kind) to base fields of arbitrary characteristic. (2) Discuss their relationship to mathematical objects of a different nature. (3) Provide a…
Given a local noetherian ring $R$ whose formal completion is integral, we introduce and study the $p$-radical closure $R^\text{prc}$. Roughly speaking, this is the largest purely inseparable $R$-subalgebra inside the formal completion…
In this short note, we give a characterization of domains satisfying Serre's condition $(\mathrm{R}_1)$ in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author (K. Yanagawa,…