What is meant by the fact that $k\langle x_1, \ldots, x_n\rangle$ denotes a free associative algebra in indeterminate $x_1, \ldots, x_n$? As the question title suggests, what is meant by the fact that $k\langle x_1, \ldots, x_n\rangle$ denotes a free associative algebra in indeterminates $x_1, \ldots, x_n$? Could anybody help me unpack this statement? Could anyone give me their intuition for working with such an object?
Thank you!
 A: Just like a polynomial algebra is spanned by monomials, the free associative algebra with indeterminates $x_1, \dots, x_n$ has a basis given by monomials of the type
$$x_{i_1} \dots x_{i_k}$$
where $k \ge 0$ and $i_1, \dots, i_k \in \{ 1, \dots, n \}$, the case $k=0$ corresponding to the empty monomial, i.e. the unit of the algebra. The product is "concatenation" of such monomials extended linearly, i.e.
$$(x_{i_1} \dots x_{i_k}) \cdot (x_{i_{k+1}} \dots x_{i_{k+l}}) = x_{i_1} \dots x_{i_{k+l}}$$
The main difference with a polynomial algebra is that $x_i x_j \neq x_j x_i$ when $i \neq j$, so the order of the indeterminates matters in a monomial. For example $x_1 x_2 - x_2 x_1$ is not the zero element. This is also sometimes called the "tensor algebra".
The name comes from the fact that given any associative algebra $A$ and any elements $a_1, \dots, a_n \in A$, there exists an algebra morphism $f : k \langle x_1, \dots, x_n \rangle \to A$ such that $f(x_i) = a_i$, and that morphism is unique. If you know category theory, this means that the functor $k \langle - \rangle$ is left adjoint to the forgetful functor from associative algebras to sets. 
