Introduction

Tensor products of algebras and modules are constructed and their universal mapping properties investigated. The tensor products of modules and those of algebras are not only treated separately, but are investigated from different perspectives. The tensor product of modules is defined externally in terms of a universal property. Whereas the tensor product C of two algebras A and B is defined internally with reference only to A, B ⊂ C in terms of C only. The latter approach is well suited to answer questions of the following type. Given an algebra C containing subalgebras A ⊂ C and B ⊂ C, when does C have the structure of a tensor product C ≅ A ⊗FB? Aside from the obvious condition that elements of A and B commute and that C is “generated” by A and B, intuitively, C is a tensor product of A and B if there are “no algebraic relations between A and B” in C. The latter vague notion is made precise by defining and using the concept of linear disjointness. Both for modules and also algebras several concrete constructions of tensor products are given.

One of the more interesting aspects of this chapter is the Universal Mapping Theorem. It gives a necessary and sufficient condition on an algebra C to be an algebra tensor product of A and B, but this condition is expressed entirely in terms of algebra homomorphisms.