  For any prime number $p$ and field $k$, we characterize the $p$-retract
rationality of an algebraic $k$-torus in terms of its character lattice. We
show that a $k$-torus is retract rational if and only if it is $p$-retract
rational for every prime $p$, and that the Noether problem for retract
rationality for a group of multiplicative type $G$ has an affirmative answer
for $G$ if and only if the Noether problem for $p$-retract rationality for $G$
have a positive answer for all $p$. For every finite set of primes $S$ we give
examples of tori that are $p$-retract rational if and only if $p\notin S$.
