Honesty (total inertness) of the embedding O_d(R) → O_d(r)

Prove that for any radii r<R in [0,+∞], the natural embedding of semifirs O_d(R) into O_d(r) is totally inert (hence honest), i.e., it preserves fullness and inner rank of matrices; equivalently, show that every square full matrix over O_d(R) remains full over O_d(r). This is motivated by extending Amitsur’s theorem to rule out meromorphic identities in M_d(R).

Background

The paper introduces the semifirs O_d(R) of uniformly analytic non-commutative (NC) functions and their universal skew fields of fractions M_d(R). A key technical property needed to interpret elements of M_d(R) as bona fide NC functions is that the embeddings O_d(R)→O_d(r) preserve inner rank (honesty/total inertness), ensuring matrices that are full over the larger ring remain full over the smaller ring.

Establishing total inertness would imply an extension of Amitsur’s theorem to M_d(R), ruling out non-trivial meromorphic identities and clarifying the functional nature of elements of M_d(R).