Structural results on lifting, orthogonality and finiteness of idempotents

TL;DR

This paper explores the properties of idempotents in rings, providing new characterizations of lifting idempotents, their orthogonality, and the finiteness of idempotents, with applications to ring spectra and primitive idempotents.

Contribution

It introduces novel results on lifting idempotents, characterizes their behavior modulo ideals, and links the number of idempotents to the structure of the spectrum, generalizing existing theories.

Findings

Lifting idempotents modulo the Jacobson radical is characterized.

The number of idempotents in a ring is finite iff it equals 2 to the power of the number of connected components.

Primitive idempotents correspond to isolated points in the prime spectrum of zero-dimensional rings.

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 $2^{\kappa}$ where $\kappa$ 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.