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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 be a subring (with unity), and suppose that has finite spectrum. Then is dense in , the topological closure of the fraction field of .*

In particular — assuming a finite spectrum — it follows that the rank of (considered as an abelian group) is bounded below by two. Unfortunately, this is as far as our previous methods will take us, even when (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 ) the fact that 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 is a subring (with unity). If has finite spectrum, then has infinite rank as an abelian group.*