Rings where a non-nilpotent sum of units is a unit

TL;DR

This paper investigates rings where sums of units are either units or nilpotent, characterizing such rings in algebraic geometry and number theory, and introducing the concept of unit dimension and the unit-additive closure.

Contribution

It provides a comprehensive characterization of unit-additive rings, explores their geometric and algebraic properties, and introduces the concept of unit dimension and the construction of the unit-additive closure.

Findings

Affine semigroup rings are unit-additive iff the base ring is and the semigroup has no invertible elements.

Coordinate rings of elliptic curves are always unit-additive.

Rings of unit dimension 1 include rings of integers of number fields, power series rings, and most local rings.

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.