  If $R$ is a regular and semiartinian ring, it is proved that the following
conditions are equivalent: (1) $R$ is unit-regular, (2) every factor ring of
$R$ is directly finite, (3) the abelian group $K_0(R)$ is free and admits a
basis which is in a canonical one to one correspondence with a set of
representatives of simple right $R$-modules. For the class of semiartinian and
unit-regular rings the canonical partial order of $K_0(R)$ is investigated and
the directed abelian groups which are realizable as $K_0(R)$ of these rings are
classified.
