Similarity classes for nilpotent operators over dedekind domains Original Research Article

Let R be a commutative Dedekind domain, and let V denote a finitely generated torsion-free module over R. Let gl(V) denote the R-module endomorphisms of V, N(V) subset of or equal to gl(V) the set of nilpotent endomorphisms, and GL(V) the automorphisms of V. We construct a canonical filtration and invariant ideals associated to elements of N(V) to study several GL(V)-invariant properties of N(V), under the similarity action (g, L)maps togLg−1, L set membership, variant N(V), g set membership, variant GL(V). We use these invariants to give a finite determinacy criterion for the similarity of nilpotent endomorphisms N(V).