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Abstract: In this article we construct characteristic elements for a certain class of Iwasawa modules in noncommutative Iwasawa theory. These elements live in the first K-group of the localisation of the Iwasawa algebra with respect to a certain Ore-Set, whose existence is one of the main problems we settle. The evaluation of the characteristic element of a module M under the Iwasawa algebra of the p-adic Lie group G is related to the (twisted) G-Euler characteristic of M. We apply these results to study the arithmetic of elliptic curves E (without CM) over a number field k. In particular, we relate the characteristic element of the Selmer group of E over the extension which arises by adjoining the p-power division points to k to the (classical) characteristic polynomial associated with the Selmer group over the cyclotomic Zp-extension. Finally we suggest (the shape of) a formulation of a noncommutative main conjecture.