# tensor algebra

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

 $\mathcal{T}(M)=\bigoplus_{n=0}^{\infty}\mathcal{T}_{n}(M)$

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

 $\mathcal{T}_{n}(M)=M^{\otimes n}=\overbrace{M\otimes\cdots\otimes M}^{n\text{ % times}},\quad n=1,2,\ldots,$

and $\mathcal{T}_{0}(M)=R$. The multiplication $m:\mathcal{T}(M)\times\mathcal{T}(M)\to\mathcal{T}(M)$ is given by the usual tensor product   :

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

## Remark 1.

One can generalize the above definition to cover the case where the ground ring $R$ is non-commutative by requiring that the module $M$ is a bimodule with $R$ acting on both the left and the right.

## Remark 2.

From the point of view of category theory     , one can describe the tensor algebra construction as a functor  $\mathcal{T}$ from the category  of $R$-module to the category of $R$-algebras that is left-adjoint to the forgetful functor   $\mathcal{F}$ from algebras to modules. Thus, for $M$ an $R$-module and $S$ an $R$-algebra, every module homomorphism  $M\to\mathcal{F}(S)$ extends to a unique algebra homomorphism $\mathcal{T}(M)\to S$.

Title tensor algebra TensorAlgebra 2013-03-22 13:17:21 2013-03-22 13:17:21 rmilson (146) rmilson (146) 13 rmilson (146) Definition msc 15A69 FreeAssociativeAlgebra tensor power