Homomorphisms of C*-algebras and their K-theory

Let $A$ and $B$ be C*-algebras and $\phi \text{\hspace{0.17em}⁣}:A\to B$ be a $\ast$-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 $\phi$ is unital and injective. We also show that $A$ is embeddable in $B$ if ${\mathrm{K}}_0(\phi )$ is injective and $A$ has stable rank one and real rank zero.