Tight closure in non-excellent rings

Some examples are constructed to demonstrate that tight closure need not satisfy several key properties when the ring is not excellent. In particular, there are counterexamples to colon-capturing and persistence. A consequence is that R+ need not be Cohen–Macaulay when R is a local ring of characteristic p even when R is universally catenarian. A new closure operation for non-excellent local rings of characteristic p is introduced and it is seen to have most of the properties associated with tight closure.