\(K\)-theory of rings with idempotents (Q1326048)

Let \(e\) be some idempotent in a Noetherian ring \(T\). The author proves that there is a long exact sequence \[ \cdots \to K_ *' (eTe/ eT(1 - e)Te) \to K_ *'(T) \to K_ *'((1 - e) T(1 - e)) \to \cdots \] for the \(K\)-theory \(K_ *'\) of finitely generated modules. Under suitable conditions this sequence splits and therefore yields a decomposition of \(K_ *' (T)\). The results are applied to endomorphism rings and graded modules.