  A key test of the supernova triggering and injection hypothesis for the
origin of the solar system's short-lived radioisotopes is to reproduce the
inferred initial abundances of these isotopes. We present here the most
detailed models to date of the shock wave triggering and injection process,
where shock waves with varied properties strike fully three dimensional,
rotating, dense cloud cores. The models are calculated with the FLASH adaptive
mesh hydrodynamics code. Three different outcomes can result: triggered
collapse leading to fragmentation into a multiple protostar system; triggered
collapse leading to a single protostar embedded in a protostellar disk; or
failure to undergo dynamic collapse. Shock wave material is injected into the
collapsing clouds through Rayleigh-Taylor fingers, resulting in initially
inhomogeneous distributions in the protostars and protostellar disks. Cloud
rotation about an axis aligned with the shock propagation direction does not
increase the injection efficiency appreciably, as the shock parameters were
chosen to be optimal for injection even in the absence of rotation. For a shock
wave from a core-collapse supernova, the dilution factors for supernova
material are in the range of $\sim 10^{-4}$ to $\sim 3 \times 10^{-4}$, in
agreement with recent laboratory estimates of the required amount of dilution
for $^{60}$Fe and $^{26}$Al. We conclude that a type II supernova remains as a
promising candidate for synthesizing the solar system's short-lived
radioisotopes shortly before their injection into the presolar cloud core by
the supernova's remnant shock wave.
