Homomorphisms of C*-algebras and their K-theory

Let $A$ and $B$ be C*-algebras and $\varphi\colon A\to B$ be a $*$-homomorphism. We discuss the properties of the kernel and (co-)image of the induced map $\mathrm{K}_{0}(\varphi)\colon \mathrm{K}_{0}(A) \to \mathrm{K}_{0}(B)$ on the level of K-theory. In particular, we are interested in the case that the co-image is torsion free, and show that it holds when $A$ and $B$ are commutative and unital, $B$ has real rank zero, and $\varphi$ is unital and injective. We also show that $A$ is embeddable in $B$ if $\mathrm{K}_{0}(\varphi)$ is injective and $A$ has stable rank one and real rank zero.