K0 of a semilocal ring

Let R be a semilocal ring, that is, R modulo its Jacobson radical J(R) is artinian. Then K0(R/J(R)) is a partially ordered abelian group with order-unit, isomorphic to (Zn, ≤, u), where ≤ denotes the componentwise order on Zn and u is an order-unit in (Zn, ≤). Moreover, the canonical projection π:R → R/J(R) induces an embedding of partially ordered abelian groups with order-unit K0(π): K0(R) → K0(R/J(R)). In this paper we prove that every embedding of partially ordered abelian groups with order-unit G → Zn can be realized as the mapping K0(π): K0(R) → K0(R/J(R)) for a suitable hereditary semilocal ring R.