  Given a partial action \alpha of a group G on an associative algebra A we
consider the crossed product A x_\alpha G. Using the algebras of multipliers of
ideals of A we prove that A x_\alpha G is associative, provided that all ideals
of A are idempotent. This generalizes a previous result on the associativity of
A x_\alpha G in the context of C*-algebras. We also give a criteria for the
existence of a global extension of a given partial action on an algebra and use
crossed products to study relations between partial actions of groups on
algebras and partial representations. As an application we endow partial group
algebras with crossed product structure.
