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Feedback classification of single-input systems over von Neumann regular rings

Andres Saez-Schwedt
This work deals with linear systems with scalars in a
commutative ring R with the property of being “von
Neumann regular”, i.e. R is zero-dimensional and has
no nonzero nilpotents. We prove that every single-input,
n-dimensional system over R is feedback equivalent to
a special normal form, whose existence actually characterizes
the class of von Neumann regular rings. This
normal form, which captures completely the structure
of the reachable submodule of the system, is associated
to a collection of n principal ideals generated by idempotent
elements f1, . . . , fn, each dividing the following
one. The normal form can be obtained by an explicit algorithm,
which is implemented in PARI-GP in the case
R = Z/(dZ), where d is a squarefree integer.
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