  We show that the Kauffman bracket skein algebra of any oriented surface F
(possibly with marked points in its boundary) has no zero divisors and that its
center is generated by knots parallel to the unmarked components of the
boundary of F. Furthermore, we show that skein algebras are Noetherian and Ore.
Our proofs rely on certain filtrations of skein algebras induced by pants
decompositions of surfaces. We prove some basic algebraic properties of the
associated graded algebras along the way.
