The star-products in the smooth deformation quantization theory invented by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer consist of formal power series of bidifferential op-erators for the algebra of smooth real or complex-valued functions on a given Poisson manifold.Hence they ‘localize’ in the analytic sense that they give rise to deformations of the the algebra ofthose smooth functions which are only defined on a given open set U of the manifold. We show thatthis notion of ‘analytic localization’ is equivalent to noncommutative ‘algebraic localization’ of thedeformed algebra with respect to the subset of those functions nowhere vanishing on U: here the latterturns out to satisfy the right (and also left) Ore condition whence the formal right algebra of fractions(algebraic localization) is isomorphic to the deformed algebra of local functions. The proofs are basedon old work by Malgrange and Tougeron on the commutative algebra of smooth function rings of the 60Õs and 70Õs. An analogous treatment also works for analytic and algebraic germs around a givenpoint. This is a joint work with Hamilton Araujo and Benedikt Hurle.