the unit are replaced by weaker axioms. The easiest example of a weak Hopf algebra is a groupoid algebra; other examples are face algebras [@Hayashi], quantum groupoids [@Ocneanu], generalized Kac algebras [@Yamanouchi] and quantum transformation groupoids [@NV]. Temperley-Lieb algebras give rise to weak Hopf algebras (see [@NV]). A purely algebraic study of weak Hopf algebras has been presented in [@BohmNSI]. A survey of weak Hopf algebras and their applications may be found in [@NV]. It has turned out that many results of classical Hopf algebra theory can be generalized to weak Hopf algebras.\
Yetter-Drinfeld modules over finite dimensional weak Hopf algebras over fields have been introduced by Nenciu [@Nenciu]. It is shown in [@Nenciu] that the category of finite dimensional Yetter-Drinfeld modules is isomorphic to the category of finite dimensional modules over the Drinfeld double, as introduced in the appendix of [@Bohm]. It is also shown that this category is braided isomorphic to the center of the category of finite dimensional $H$-modules. In this note, we discuss Yetter-Drinfeld modules over weak bialgebras over commutative rings. The results in [@Nenciu] are slightly generalized and more properties are given.\
In [Section \[se:2\]]{}, we compute the weak center of the category of modules 