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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.

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