tensor algebra

Let $R$ be a commutative ring, and $M$ an $R$ -module. The tensor algebra$$ \mc{T}(M) = \bigoplus_{n=0}^\infty \mc{T}_n(M$$ is the graded $R$ -algebra with $n^{th}$ graded component simply the $n^{th}$ tensor power:$$ \mc{T}_n(M) = M^{\otimes n} =\overbrace{M\otimes \cdots \otimes M}^{n\text{ times}},\quad n=1,2,\ldots$$ and $\mc{T}_0(M)=R$ . The multiplication $m:\mc{T}(M)\times \mc{T}(M)\to\mc{T}(M)$ 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 $\mc{T}$ from the category of $R$ -module to the category of $R$ -algebras that is left-adjoint to the forgetful functor $\mc{F}$ from algebras to modules. Thus, for $M$ an $R$ -module and $S$ an $R$ -algebra, every module homomorphism $M\to \mc{F}(S)$ extends to a unique algebra homomorphism $\mc{T}(M)\to S$ .