  An equidimensional local ring is F-rational if and only if one ideal
generated by a system of parameters is tightly closed. The question of whether
a non-equidimensional local ring can have a tightly closed ideal generated by a
system of parameters has been a long-standing open problem, and for certain
classes of non-equidimensional rings we prove that this is not possible. A key
point is that tight closure has a colon capturing property in equidimensional
rings that it does not have in non-equidimensional rings. We define a new
closure operation, one that rectifies the absence of the colon capturing
property of tight closure in non-equidimensional rings. This closure operation
agrees with tight closure when the ring is equidimensional, and we prove that
the F-rationality of a local ring is equivalent to a single system of
parameters being closed with respect to this new closure operation.
