Minimal Prüfer-Dress rings and products of idempotent matrices

Laura Cossu; Paolo Zanardo

Home/Archive/arXiv

arXiv · 1811.09092

Minimal Prüfer-Dress rings and products of idempotent matrices

Laura Cossu·Paolo Zanardo

View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate a special class of Prüfer domains, firstly introduced by Dress in 1965. The {\it minimal Dress ring} $D_K$, of a field $K$, is the smallest subring of $K$ that contains every element of the form $1/(1+x^2)$, with $x\in K$. We show that, for some choices of $K$, $D_K$ may be a valuation domain, or, more generally, a Bézout domain admitting a weak algorithm. Then we focus on the minimal Dress ring $D$ of $\mathbb{R}(X)$: we describe its elements, we prove that it is a Dedekind domain and we characterize its non-principal ideals. Moreover, we study the products of $2\times 2$ idempotent matrices over $D$, a subject of particular interest for Prüfer non-Bézout domains.

Explore related subjects

math.AC math.RA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Laura Cossu, Paolo Zanardo. 2018-11-22. Minimal Prüfer-Dress rings and products of idempotent matrices. https://arxiv.org/abs/1811.09092

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Bourbaki Degree of Line Arrangements

We study the Bourbaki degree of line arrangements in the projective plane and its interaction with the intersection lattice and the syzygies of the gradient ideal. We show that the Bourbaki degree is obtained by evaluating the reduced characteristic polynomial at the initial degree of the first syzygy module. This leads to a sharp upper bound and to refinements involving the two largest intersection multiplicities, with consequences for the global Tjurina number, the freeness defect, and Terao's conjecture, including an improvement of Dimca's numerical criterion when the smaller exponent is at least five. We also establish addition--deletion formulas and show that the defect from the sharp bound is monotone under line addition. Finally, we prove that the Bourbaki degree is determined by the intersection lattice for arrangements with at most eight lines, while examples with isomorphic intersection lattices and different Bourbaki degrees exist for every number of lines at least nine.

math.AC↗

arXiv2026

On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action

T. Kambayashi had shown that $\mathbb{A}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable $\mathbb{A}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for $\mathbb{A}^2$-forms over arbitrary field extensions admitting a nontrivial $\mathbb{G}_a$-action. From this structure theorem we derive some conditions under which an $\mathbb{A}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial $\mathbb{A}^2$-form having a $k$-rational point and a non-trivial $\mathbb{G}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field.

math.AC↗

arXiv2026

Symbolic powers of the ideal of$n$ general points in $P^{n-1}$

Problem L of Fröberg--Lundqvist--Oneto--Shapiro asks for the difference between the Hilbert series of ordinary and symbolic powers of the ideal of general points in projective space. We solve this completely for $n$ general points of $\PP^{n-1}$. Besides a closed formula for \[ \HS(S/I^m)-\HS(S/I^{(m)}), \] we determine all minimal monomial generators of $I^{(m)}$, and describe the symbolic Rees algebra. We also show that containment $I^{(m)}\subseteq I^r$ is detected solely by initial degrees. This gives the exact containment threshold, the Waldschmidt constant $\walpha$, the resurgence $\Res$, and the asymptotic resurgence $\aRes$: \[ \walpha(I)=\frac{n}{n-1}, \qquad \Res(I)=\aRes(I)=\frac{2(n-1)}{n}. \] We also take the first step beyond $n$ points: for $n+1$ general points of $\PP^{n-1}$ --- again a rigid, non-monomial configuration --- we identify the defining quadrics, resolve the case $n=3$ completely (a complete intersection, with $J^{(m)}=J^m$ for all $m$ and resurgence $1$), and propose an exact Waldschmidt-constant formula $\walpha=\frac{n+1}{n-1}$ for all $n$, verified computationally in every case we could check.

math.AC↗