  The rings whose simple right modules are absolutely pure are called right
$SAP$-rings. We give a new characterization of right $SAP$ rings, right $V$
rings, and von Neumann regular rings. We also obtain a new decomposition theory
of right selfinjective von Neumann regular rings. The relationships between
$SAP$-rings, $V$-rings, and von Neumann regular rings are explored. Some recent
results obtained by Faith are generalized and the results of Wu-Xia are
strengthened.
