Algebraic K-theory of endomorphism rings

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let $\mathcal{C}$ be an additive category, and let $Y\text{ra}X$ be a covariant morphism of objects in $\mathcal{C}$. Then $K_n\big({\mathcal C}(X\oplus Y)\big)\simeq K_n\big({{\mathcal C},Y}(X)\big)\oplus K_n\big({\mathcal C}(Y)\big)$forall$1\le n\in $\mathbb{N}$$,where${{\mathcal C},Y}(X)$ is the quotient ring of the endomorphism ring ${}_{\mathcal{C}}(X)$ of $X$ modulo the ideal generated by all those endomorphisms of $X$ which factorize through $Y$. Moreover, let $R$ be a ring with identity, and let $e$ be an idempotent element in $R$. If $J:=ReR$ is homological and ${}_{R}J$ has a finite projective resolution by finitely generated projective $R$-modules, then $K_n®\simeq K_n(R/J)\oplus K_n(eRe)$forall$n\in $\mathbb{N}$$. This reduces calculations of the higher algebraic $K$-groups of $R$ to those of the quotient ring $R\mathrm{/}J$ and the corner ring $eRe$, and can be applied to a large variety of rings: Standardly stratified rings, hereditary orders, affine cellular algebras and extended affine Hecke algebras of type $\tilde{A}$.