  Tracial Rokhlin property was introduced by Phillips to prove various
structure theorems for crossed product. But it is defined for actions on simple
C*-algebras only. This paper consists of two major parts: In section 2 and 3,
we study the permanence properties and give a complete classification of
tracial Rokhlin property for product-type actions; In section 4 and 5, we
introduce the weak tracial Rokhlin property for actions on non-simple
C*-algebras. We prove that when the action has the weak tracial Rokhlin
property and the crossed product is simple, the properties on $A$ of having
tracial rank $\leq k$, or real rank 0, or stable rank 1, can be inherited by
the crossed product.
