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This post serves to resolve some questions posed at the end of a previous article, Unit-Prime Dichotomy for Complex Subrings, in which we began to look at the tension between the relative sizes of the units group and the spectrum of  a complex subring (with unity).  As a disclaimer, I assume from this point familiarity with the topics presented therein.  To begin, we recall a Theorem from the prequel (Theorem 4):

Theorem: Let $R \subset \mathbb{C}$ be a subring (with unity), and suppose that $R$ has finite spectrum.  Then $R^\times$ is dense in $\mathrm{cl}(\mathrm{Frac}(R))$, the topological closure of the fraction field of $R$.

In particular — assuming a finite spectrum — it follows that the rank of $R^\times$ (considered as an abelian group) is bounded below by two. Unfortunately, this is as far as our previous methods will take us, even when $R \not\subset \mathbb{R}$ (see the Exercises).

The primary objective of this post is an extension to this result.  Specifically, we would like to capture in a purely algebraic way (e.g. without mention of the topology on $\mathbb{C}$) the fact that $R^\times$ becomes quite large in certain rings with finite spectrum.  After the fold we’ll accomplish exactly that, with the following Theorem and its generalizations to a wide class of commutative rings:

Theorem 1: Suppose that $R \subset \mathbb{C}$ is a subring (with unity).  If $R$ has finite spectrum, then $R^\times$ has infinite rank as an abelian group.