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This paper is a continuation of our previous work [10]. By GAERS-1, we denote the class of generalized abelian exchange rings with stable range 1. In this paper, we first prove that for any ring R ∈ GAERS-1 and any ideal I of R, K0(R/I) is an archimedean ℓ-group, which is a natural generalization of [10, Theorem 5.3]. As applications, we establish explicit characterizations for the K0-simplicity of such rings in the sense of [3], and investigate the norm completeness of their K0-groups. Finally, we characterize the primitive idempotents in R by K0(R) with ordered structure, from which we can further determine completely the structure of K0(R). [ABSTRACT FROM AUTHOR]