  For a twisted partial action \Theta of a group G on an (associative
non-necessarily unital) algebra A over a commutative unital ring k, the crossed
product A X_\Theta G is proved to be associative. Given a G-graded k-algebra B
= \oplus_{g\in G}\B_g with the mild restriction of homogeneous non-degeneracy,
a criteria is established for B to be isomorphic to the crossed product B_1
X_\Theta G for some twisted partial action of G on B_1. The equality
B_g\B_{g^{-1}}B_g = \B_g for all g\in G is one of the ingredients of the
criteria, and if it holds and, moreover, B has enough local units, then it is
shown that B is stably isomorphic to a crossed product by a twisted partial
action of G.
