ABSTRACT. We introduce and study the definition, main properties and applications of iterated twisted tensor products of algebras, motivated by the problem of defining a suitable representative for the product of spaces in noncommutative geometry. We find conditions for constructing an iterated product of three factors, and prove that they are enough for building an iterated product of any number of factors. As an example of the geometrical aspects of our construction, we show how to construct differential forms and involutions on iterated products starting from the corresponding structures on the factors, and give some examples of algebras that can be described within our theory. We prove a certain result (called "invariance under twisting") for a twisted tensor product of two algebras, stating that the twisted tensor product does not change when we apply certain kind of deformation. Under certain conditions, this invariance can be iterated, containing as particular cases a number of independent and previously unrelated results from Hopf algebra theory.

No abstract

The first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module M produce these decompositions: the lattice decompositions. In a firstétage this can be done using endomorphisms of M , which produce a decomposition of the ring End R (M ) as a product of rings, i.e., they are central idempotent endomorphisms. But since not every central idempotent endomorphism produces a lattice decomposition, the classical theory is not of application. In a second step we characterize when a particular module M has a lattice decomposition; this can be done, in the commutative case in a simple way using the support, Supp(M ), of M ; but, in general, it is not so easy. Once we know when a module decomposes, we look for characterizing its decompositions. We show that a good framework for this study, and its generalizations, could be provided by the category σ[M ], the smallest Grothendieck subcategory of Mod − R containing M .

No abstract