Susan's question When can number rings be spanned (as $\mathbb{Z}$-modules) by units? smells like an algebraic K-theory question in disguise. I'll reformulate the question first:

Given an integral domain (for safety) $R$, let $A$ be the smallest subring containing the units $R^\times$. When does $A = R$? Examples: any field, the integers, the Gaussian or Eisenstein integers, an easily described class of real quadratic rings.

Consider the related high-brow property of an integral domain $R$:

(P): For all (sufficiently large?) $N$, and all partitions $n_1 + \cdots + n_t = N$, the group $GL_N(R)$ is generated by $GL_N(Z)$ and $\prod_{i=1}^t GL_{n_i}(R)$.

There are definitely better ways of writing property (P), but my point is that property (P) seems very much like an asymptotic statement about the structure theory of $GL_N(R)$ as a group containing $GL_N(Z)$.

If $A = R$, then I think $R$ has property (P). So two questions:

Is property (P) equivalent to $A = R$? I'd guess yes. Edit: maybe if $K_1(R) = R^\times$?

Can a K-theory expert reformulate property (P) in terms of the algebraic K-theory of $R$?

At a fundamental level, does the property $A = R$ depend only on the algebraic K-theory of $R$? Does Milnor K-theory come into play (e.g. since $R^\times$ might not equal $K_1(R)$ if $R$ is not a Euclidean domain)?