Rings where a non-nilpotent sum of units is a unit
Neil Epstein, Jay Shapiro
TL;DR
This work introduces and develops the concept of unit-additive rings, where sums of units are constrained to lie in the unit group or the nilradical. It establishes a network of equivalences and investigates how unit-additivity behaves under monoid algebras, affine algebras, and elliptic-curve coordinate rings, linking algebraic properties to geometric notions such as maps to the punctured line and polynomial roots. A new invariant, unit dimension, measures how far a ring is from being unit-additive, and is shown to satisfy udim$\le$dim for affine algebras, with a construction showing all finite udim values occur and examples of infinite udim. The paper also defines the unit-additive closure, a universal localization capturing the smallest unit-additive overring and exploring its relationship to fraction fields and localizations. Together, these results connect classical structures (PIDs, Euclidean domains, UU rings) to unit-additivity and raise several open questions for further exploration in algebraic geometry and commutative algebra.
Abstract
A ring is *unit-additive* if a sum of units is always either a unit or nilpotent. For example, $k[X]$ and $k[X]/(X^2)$ are unit-additive, but $\mathbb Z$ is not. We prove a wide-ranging theorem about unit-additivity in semigroup rings, showing among other things that an affine semigroup ring $A[M]$ is unit-additive if and only if $A$ is unit-additive and $M$ has no nontrivial invertible elements. Passing to algebraic geometry, we show that an irreducible affine variety $V$ over an algebraically closed field $k$ has unit-additive coordinate ring if and only if any polynomial mapping $V \rightarrow k$ has a root. This then places $\mathbb A^1_k$ into the class of varieties that satisfy a version of the Fundamental Theorem of Algebra. Specializing to elliptic curves, we show that the affine coordinate ring of an elliptic curve is always unit-additive. The concept of unit additivity leads to the related concept of unit dimension -- i.e. how far is an integral domain from being unit-additive? It turns out that rings of unit dimension 1 are of some interest, as they include the rings of integers of number fields, all power series rings, and most local rings. We construct rings of all unit dimensions and show that in the affine setting, unit dimension is bounded above by Krull dimension. We also construct the *unit-additive closure* of an integral domain $D$, being the smallest subring of the fraction field of $D$ that is unit-additive, as a localization at a certain multiplicative set in $D$. Throughout, we make connections with well-studied structures like PIDs, Euclidean domains, and the UU property.