# The tensor algebra

In this post I will define the tensor algebra over a module and show that it satisfies a universal property which makes it the universal algebra with respect to algebras containing an homomorphic image of that module.

Let be an -module for a commutative and unitary ring. For each define

where denotes tensor product. Note that this definition is unambiguous considering the fact that taking tensor products is an associative construction. For define Define the tensor algebra to be

Recall that the infinite direct sum of a family of modules is the subset of the infinite product such that all but finitely many of the ‘s are For all the elements of are called k-tensors. Thus, every element of can be viewed as a finite linear combination of k-tensors for various Identifying with we can view as a submodule of

Now, defining addition componentwise in gives it the structure of an -module. To get an -algebra structure, we define the multiplication in as

and extend it by linearity and distributivity to all of

Proposition:This multiplication is well-defined and makes into an -algebra.

**Proof:** By the proof of the associativity of tensor products, there exists a unique isomorphism such that

Thus the multiplication map above is well defined. Indeed, suppose Then

QED

In fact, since , the -algebra is a graded algebra. Here is the universal property satisfied by the tensor algebra:

Theorem (Universal property):Let be any -algebra, an -module. If is any -module homomorphism, then there is a unique -algebra homomorphism such that

**Proof:** For every the map defined by is easily seen to be k-multilinear with respect to Thus for every there exists a unique -module homomorphism such that With the definition of the multiplication in it is easy to see that the induced map is an -algebra homomorphism satisfying the desired properties.

QED

**Reference:** D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.

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