Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1806.03599

Structural results on lifting, orthogonality and finiteness of idempotents

Abolfazl Tarizadeh·Pramod K. Sharma
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized in terms of certain connected sets related to that ideal. Then as an application, we obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that lifting idempotents preserves the orthogonality in countable cases. The lifting property of an arbitrary morphism of rings is characterized. As another major result, it is proved that the number of idempotents of a ring $R$ is finite if and only if it is of the form $2^κ$ where $κ$ is the cardinal of the connected components of Spec$(R)$. Finally, it is proved that the primitive idempotents of a zero dimensional ring are in 1-1 correspondence with the isolated points of its prime spectrum. These results either generalize or improve several important results in the literature.

Explore related subjects

math.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Abolfazl Tarizadeh, Pramod K. Sharma. 2021-12-28. Structural results on lifting, orthogonality and finiteness of idempotents. https://doi.org/10.1007/s13398-021-01199-w

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↗