Topic
Characteristic
About: Characteristic is a research topic. Over the lifetime, 51 publications have been published within this topic receiving 1011 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, it was shown that if a ring of characteristic zero has rational singularity, then its modulo p reduction is F -rational for almost all characteristic p. This result leads to the correspondence of F -regular rings and log terminal singularities.
Abstract: The notions of F -rational and F -regular rings are defined via tight closure, which is a closure operation for ideals in a commutative ring of positive characteristic. The geometric significance of these notions has persisted, and K. E. Smith proved that F -rational rings have rational singularities. We now ask about the converse implication. The answer to this question is yes and no. For a fixed positive characteristic, there is a rational singularity which is not F -rational, so the answer is no. In this paper, however, we aim to show that the answer is yes in the following sense: If a ring of characteristic zero has rational singularity, then its modulo p reduction is F -rational for almost all characteristic p . This result leads us to the correspondence of F -regular rings and log terminal singularities.
252 citations
TL;DR: In this paper, it was shown that any nonzero ideal of a semiprime ring with polynomial identity intersects the center nontrivially (Theorem 2).
Abstract: Introduction. The purpose of this paper is to provide a fresh outlook to various questions on rings with polynomial identity by examining the centers of such rings. This approach yields the interesting result that any nonzero ideal of a semiprime ring with polynomial identity intersects the center nontrivially (Theorem 2). There are at least two interesting consequences to Theorem 2: a generalization of Wedderburn's theorem (any semiprimitive ring with polynomial identity, whose center is a field, is simple) and a strengthening of Posner's theorem [1] (any prime ring with a polynomial identity has a simple ring of quotients whose center is the quotient field of the center of the prime ring). The proofs are elementary modulo Jacobson [3]. Of course rings are not necessarily commutative and for the sake of simplicity we assume a unit 1. The key argument in this paper is an application of Formanek's central polynomials for matrix algebras over a field, whose important properties are [2] : Let Mn be an n x n matrix algebra over an arbitrary field. Then there exists a polynomial gn(Xl9.. .,Xm) which has coefficients in Z; is homogeneous (degree > 0) in every variable and linear in all but the first variable; takes values in the center for every specialization in Mn; and is nonvanishing for some specialization.
187 citations
TL;DR: In this article, it was shown that a graded subring of a polynomial ring over a perfect field of characteristic splits is a simple ring, and that this is true in prime characteristic.
Abstract: Let be a finite dimensional representation of a linearly
reductive group . Motivated by their work on classical
rings of invariants, Levasseur and Stafford asked whether the
ring of invariants under has a simple ring
of differential operators.In this paper, we show that this is true in prime characteristic.
Indeed, if is a graded subring of a polynomial ring over a perfect field
of characteristic splits, then is a simple ring. In the last section of the paper, we discuss how
one might try to deduce the characteristic zero case from this result.
As yet, however, this is a subtle problem and the answer to the question of
Levasseur and Stafford remains open in characteristic zero.http://www.luc.ac.be/Research/Algebra1991 Mathematics Subject Classification: 16S32, 16G60, 13A35.
179 citations
58 citations
TL;DR: In this article, the authors proved that the F-signature of an affine semigroup ring of positive characteristic is always a rational number, and described a method for computing this number.
45 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2018 | 2 |
| 2017 | 2 |
| 2015 | 1 |
| 2012 | 3 |
| 2011 | 3 |