  In 1971, J. Lipman introduced the notion of strict closure of a ring in
another, and established the underlying theory in connection with a conjecture
of O. Zariski. In this paper, for further developments of the theory, we
investigate three different topics related to strict closure of rings. The
first one concerns construction of the closure, and the second one is the study
regarding the question of whether the strict closedness is inherited under flat
homomorphisms. We finally handle the question of when the Arf closure coincides
with the strict closure. Examples are explored to illustrate our theorems.
