bibkey: minacnguyentan2026sums
authors: Ján Mináč, Tung T. Nguyen, Nguyen Duy Tân
year: 2026
title: Sums of units in finite rings and applications to Cayley graphs
doi: null
url: https://arxiv.org/abs/2607.00404v1
claim: Question 3.5 asks whether the exact n-sum property of a finite field transfers to every finite group algebra with units restricted by standard augmentation.
strata_touched:
  - D5/S3/ArithUnits/MinacNguyenTanQuestion35
license: CC-BY-4.0
triage: anchor

Sums of units in finite rings

Verified locator

The versioned record is https://arxiv.org/abs/2607.00404v1. The primary question is https://arxiv.org/html/2607.00404v1#S3.Thmthm5. Definition 3.1 defines an exact number of summands, with repetitions allowed. The opening of Section 3 assumes a subgroup of units containing minus one. Section 3.2 assumes a finite field of cardinality greater than two and at least two summands, and defines the coefficient-sum augmentation on the actual group algebra.

Question and scope

Suppose that F has the n-sum property with respect to U. Is it true that F[G] also has the n-sum property with respect to U_{ε_G}?

Here G is an arbitrary finite group, U is a subgroup of F’s multiplicative units containing minus one, and U_{ε_G} consists of genuine group-algebra units whose standard augmentation lies in U. The number of summands remains n. Remark 3.6’s n times the group order is a different, weaker conclusion. Proposition 3.7 and Corollary 3.9 cover the modular p-group and nonmodular cases, respectively; they do not settle the arbitrary-group question.

Mathematical correspondence

The formal source uses MonoidAlgebra F G, its actual unit type, and MonoidAlgebra.lift of the constant-one group homomorphism. This lift equals the finite sum of coefficients. augmentationUnitSubgroup is the preimage of U under the induced homomorphism of genuine unit groups. HasNSumProperty S n quantifies over every ring element and a function on Fin n whose values lie in S and whose sum equals that element.

The proof descends to the semisimple Jacobson quotient and uses its actual Wedderburn matrix product. Orthogonal factor idempotents identify the factor selected by augmentation. The selected simple factor injects into F, so its entries are the prescribed scalar sums. Finite division rings are fields; rank normal form and a scalar distinct from zero and one construct two units in every remaining factor. Additional unit summands extend this to the original n. Quotient-unit lifts remain units in the finite ring, and the sum error lies in the radical and has zero augmentation. Adding that error to one lift makes the sum exact without altering any augmentation.

Literature boundary

The versioned source explicitly leaves the general case as Question 3.5. The bounded exact-title, question-number, prescribed-augmentation and repository search recorded in preregistration issue 15666 found no inspected prior resolution. This is a bound on the inspected sources, not a claim of worldwide absence. Classical matrix decomposition and radical lifting supply methods; the source’s Theorem 2.1 is not assumed as a Lean axiom.