Witt vectors with coefficients and characteristic polynomials over non-commutative rings

Abstract For a not-necessarily commutative ring $R$ we define an abelian group $W(R;M)$ of Witt vectors with coefficients in -bimodule $M$ . These groups generalize the usual big rings and prove that they have analogous formal properties structure. One main result is $W(R) := W(R;R)$ Morita invariant -linear endomorphism $f$ finitely generated projective -module characteristic element $\chi _f \in W(R)$ This non-commutative analogue classical polynomial show it has similar properties. The assignment $f \mapsto \chi _f$ induces isomorphism between suitable completion cyclic $K$ -theory $K_0^{\mathrm {cyc}}(R)$ $W(R)$