Iterated crossed products

We define a “mirror version” of Brzezinski’s crossed product and we prove that, under certain circumstances, a Brzezinski crossed product D\otimes_{R, \sigma}V and a mirror version W\bar{\otimes}_{P, \nu}D may be iterated, obtaining an algebra structure on W\otimes D\otimes V. Particular cases of this construction are the iterated twisted tensor product of algebras and the quasi-Hopf two-sided smash product.