In this paper, a su cient condition is given under which the smash product A#H is a trans nite left free normalizing extension of an algebra A. Moreover, the result is applied to a skew semigroup ring, a skew group ring and the quantum group Uq(sl(2)) such that some properties are shown.

We investigate a generalization of Hopf algebra slq (2) by weakening the invertibility of the generator K, i.e. exchanging its invertibility KK−1=1 to the regularity KKK=K. This leads to a weak Hopf algebra wslq (2) and a J -weak Hopf algebra vslq (2) which are studied in detail. It is shown that the monoids of group-like elements of wslq (2) and vslq (2) are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from wslq (2) a quasi-braided weak Hopf algebra U w q is constructed and it is shown that the corresponding quasi-R-matrix is regular RwRˆwRw =Rw.

q-fundamental matrices are introduced and studied. Elementary operations on quantum matrices are discussed. The q-generalization of the classical Laplace's theorem is found, An application of the result is given.