bibkey: sontag1980generalized authors: “Eduardo D. Sontag” year: 1980 title: “On Generalized Inverses of Polynomial and Other Matrices” doi: null url: https://www.sontaglab.org/FTPDIR/wgi.pdf claim: “For maps between fixed finitely generated projective modules over an integral domain, weak generalized inverse existence is equivalent to a split image and constant rank over every field algebra.” strata_touched:

• D5/S3/HomologicalAlgebra/IntegerMatrixInnerInverse license: citation-only triage: anchor

Splitting and maximal-minor ideals

Sontag, IEEE Transactions on Automatic Control, AC-25(3), June 1980, printed page 515, Theorem 2 and Remark 1. The modules are fixed finitely generated projective modules, and the coefficient ring is an integral domain. An $R$-field means a field equipped with an $R$-algebra structure. The theorem equates weak generalized inverse existence, image splitting as a direct summand of the target, and rank independent of the $R$-field. The weak generalized inverse in the paper satisfies both $ABA=A$ and $BAB=B$.

Remark 1 states the matrix criterion using the ideal generated by all rank-sized minors. Over a PID it is equivalent to their gcd being a unit, or to the nonzero Smith invariant factors being units. Thus a single maximal nonzero unit minor is sufficient. A free abstract image is not the same as a split embedded image: the image of $[2]$ is free, but its inclusion into the integers does not split and there is no integer inner inverse.

The repository theorem constructs an inner inverse directly from a maximal minor of a finite TU integer matrix. It does not formalize Sontag’s general equivalence, arbitrary projective-module theory, the relative-homology criterion, or chain contractibility. Total unimodularity is sufficient here, not necessary: an integral invertible matrix with an entry $2$ has an integral inverse but is not TU.

Verified locator

• https://www.sontaglab.org/FTPDIR/wgi.pdf, original scan, printed page 515, Theorem 2 and Remark 1.