I've got a paper which shows that when the spectral theorem (as a statement that every self-adjoint matrix can be unitarily diagonalised) is naively generalised to $*$-algebras other than the complex numbers, matrix decompositions like SVD, Takagi, skew-Takagi (and analogues of these over the quaternions) all get unified. In particular, a pair consisting of a left singular vector and a right singular vector behaves like an eigenvector in its own right.
The idea uses change-of-rings quite heavily. Ring extension is used to unify matrix decomposition, and ring restriction is used to pass from non-commutative to commutative rings, to ease calculations.
Is there a place where this would be of interest, and is publishable? There's a chance that this question is not appropriate here.