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It is proved that feedback classification of a linear system over acommutative von Neumann regular ring R can be reduced to the classification ofa finite family of systems, each of which is properly split into a reachableand a non-reachable part, where the reachable part is in a Brunovski-typecanonical form, while the non-reachable part can only be altered by similarity.If a canonical form is known for similarity of matrices over R, then it can beused to construct a canonical form for feedback equivalence. An explicitalgorithm is given to obtain the canonical form in a computable contexttogether with an example over a finite ring.

Feedback classification of linear systems over von Neumann regular rings