Free associative algebra

The algebra $k\langle X \rangle$ of polynomials over a field $k$ in non-commuting variables in $X$. The following universal property determines the algebra $k\langle X \rangle$ uniquely up to an isomorphism: There is a mapping $i : k \rightarrow k\langle X \rangle$ such that any mapping from $X$ into an associative algebra $A$ with a unit over $k$ can be factored through $k\langle X \rangle$ in a unique way. The basic properties of $k\langle X \rangle$ are:

1) $k\langle X \rangle$ can be imbedded in a skew-field (the Mal'tsev–Neumann theorem);

2) $k\langle X \rangle$ has a weak division algorithm, that is, the relation $$ d \left({ \sum_{i=1}^n a_i b_i }\right) < \max_i \{ d(a_i) + d(b_i) \} $$ where $a_i, b_i \in k\langle X \rangle$, all the $a_i$ are non-zero ($i = 1,\ldots,n$), $d(a_1) \le \cdots \le d(a_n)$, always implies that there are an integer $r$, $1 < r \le n$, and elements $c_,\ldots,c_{r-1}$ such that $$ d\left({ a_r - \sum_{i=1}^{r-1} a_i c_i }\right) < d(a_r) $$ and $$ d(a_i) + d(c_i) < d(a_r),\ \ \ i=1,\ldots,r-1 $$ (here $d(a)$ is the usual degree of a polynomial $a \in k\langle X \rangle$, $d(0) = -\infty$);

3) $k\langle X \rangle$ is a left (respectively, right) free ideal ring (that is, any left (respectively, right) ideal of $k\langle X \rangle$ is a free module of uniquely determined rank);

4) the centralizer of any non-scalar element of $k\langle X \rangle$ (that is, the set of elements that commute with a given element) is isomorphic to the algebra of polynomials over $k$ in a single variable (Bergman's theorem).

References

[1]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[2]	P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)