tensor algebra

Let $R$ be a commutative ring, and $M$ an $R$-module.The tensor algebra

$$𝒯(M)=\underset{n=0}{\overset{\mathrm{\infty }}{\oplus }}{𝒯}_n(M)$$

is the graded $R$-algebra with $n^{th}$graded component simply the $n^{th}$ tensor power:

$${𝒯}_n(M)=M^{\otimes n}=\stackrel{n\text{ times}}{\overbrace{M\otimes \mathrm{\cdots }\otimes M}},n=1,2,\mathrm{\dots },$$

and ${𝒯}_0(M)=R$.The multiplication $m:𝒯(M)\times 𝒯(M)\to 𝒯(M)$ is givenby the usual tensor product:

$$m(a,b)=a\otimes b,a\in M^{\otimes n},b\in M^{\otimes m}.$$

Remark 1.

One can generalize the above definition tocover the case where the ground ring $R$ is non-commutative byrequiring that the module $M$ is a bimodule with $R$ acting on boththe left and the right.

Remark 2.

From the point of view of category theory, onecan describe the tensor algebra construction as a functor $𝒯$from the category of $R$-module to the category of $R$-algebras thatis left-adjoint to the forgetful functor $ℱ$ from algebras tomodules. Thus, for $M$ an $R$-module and $S$ an $R$-algebra, everymodule homomorphism $M\to ℱ(S)$ extends to a unique algebrahomomorphism $𝒯(M)\to S$.