  We propose an exact expression for the unintegrated form of the star gauge
invariant axial anomaly in an arbitrary even dimensional gauge theory. The
proposal is based on the inverse Seiberg-Witten map and identities related to
it, obtained earlier by comparing Ramond-Ramond couplings in different
decsriptions. The integrated anomalies are expressed in terms of a simplified
version of the Elliott formula involving the noncommutative Chern character.
These anomalies, under the Seiberg-Witten transformation, reduce to the
ordinary axial anomalies. Compatibility with existing results of anomalies in
noncommutative theories is established.
