General remarks

In Section (1.3) we defined the tensor powers of an R-module M. In this chapter we shall show how the different tensor powers can be fitted together to produce an R-algebra. This algebra, which is known as the tensor algebra of M, contains M as a submodule. In fact the tensor algebra solves a certain universal problem concerned with algebras that contain a homomorphic image of M. The tensor algebra is particularly important in our context because it has, as homomorphic images, the exterior and symmetric algebras which we shall study later.

As in previous chapters R and S always denote commutative rings. All algebras are understood to be associative, rings and algebras are assumed to have identity elements, and homomorphisms of rings and algebras are required to preserve identity elements. Finally, we shall often omit the suffix from the tensor symbol ⊗ when the underlying ring can easily be inferred from the context.

The tensor algebra

Let M be an R-module. We shall define its tensor algebra in terms of a certain universal problem. To this end suppose that A is an R-algebra and that φ: M → A is a homomorphism of R-modules. If now h: A → B is a homomorphism of R-algebras, then, of course, h ∘ φ: M → B is an R-module homomorphism of M into B. This observation leads us to pose the following universal problem.