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> [!definition] Definition. ([[tensor algebra]])
> Let $M$ be a [[module]] over a [[commutative ring]] $R$. The **tensor algebra** of $M$ is the [[graded algebra|graded]] $R$-[[algebra]] $\mathbb{T}_{R}^{\bullet}(M):= \bigoplus_{ \ell \geq 0}\mathbb{T}_{R}^{\ell}(M),$
where multiplication is defined on [[tensor product of modules|pure tensors]] by $\begin{align}
M^{\otimes i} \times M^{\otimes j} & \to M^{\otimes (i+j)} \\
(m_{1} \otimes \dots \otimes m_{i}) \cdot (n_{1} \otimes \dots \otimes n_{j})&:= m_{1} \otimes \dots \otimes m_{i} \otimes n_{1} \otimes \dots \otimes n_{j}
\end{align}$
and extending by linearity.
>
Here, $\mathbb{T}_{R}^{\ell}(M)=M^{\otimes \ell}$ denotes the $\ell$th [[tensor product of modules|tensor power]] of $M$.
^definition
> [!intuition]
> Suppose $V$ is a $d$-dimensional [[vector space]] with basis $\{ v_{1},\dots,v_{d} \}$. Then $V^{\oplus n}$ is $d^{n}$-dimensional, with basis $\{ v_{i_{1}} \otimes \cdots \otimes v_{i_{n}}: ^{i_{1}=1,\dots, d}_{i_{n}=^{^{^{\vdots}}}1,\dots, d} \}.$
Thus, $V^{\oplus n}$ is like '[[homogeneous polynomial|homogeneous polynomials of degree]] $n