Strong shift equivalence and algebraic K-theory

For a semiring R, the relations of shift equivalence over R (SE-R) and strong shift equivalence over R (SSE-R) are natural equivalence relations on square matrices over R, important for symbolic dynamics. When R is a ring, we prove that the refinement of SE-R by SSE-R, in the SE-R class of a matrix A, is classified by the quotient NK₁(R)=E(A, R) of the algebraic K-theory group NK₁(R) Here (E,A) R is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over R that the refinement of its SE-R class into SSE-R classes corresponds precisely to the refinement of the GL (R[t]) equivalence class of I -tA into El (R[t]) equivalence classes. We then show this refinement is in bijective correspondence with NK₁(R)/E(A, R). For a general ring R and A invertible, the proof that E(A, R) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For R commutative, we show _AE(A, R) = NSK₁(R) the proof rests on Nenashev?s presentation of K₁ of an exact category.