K-0 of a semilocal ring

Let R be a semilocal ring, that is, R module its Jacobson radical J(R) is a rtinian. Then K-0(R/J(R)) is a partially ordered abelian group with order-u nit, isomorphic to (Z(n), less than or equal to, u), where less than or equ al to denotes the componentwise order on Z(n) and u is an order-unit in (Z( n), less than or equal to). Moreover, the canonical projection pi:R --> R/J (R) induces an embedding of partially ordered abelian groups with order-uni t K-0(pi):K-0(R) --> K-0(R/J(R)). In this paper we prove that every embeddi ng of partially ordered abelian groups with order-unit G --> Z(n) can be re alized as the mapping K-0(pi):K-0(R) --> K-0(R/J(R)) for a suitable heredit ary semilocal ring R. (C) 2000 Academic Press.