  In this note, we use the theory of test ideals and Cartier algebras to
examine the interplay between the tight and integral closures in a local ring
of positive characteristic. Using work of Schwede, we prove the abundance of
strong test ideals, recovering some older fundamental results, and use this
approach in concrete computations. In the second part of the paper, the case of
Stanley-Reisner rings is fully examined.
