Exercise 3, § 10

Let K be a field, commutative or otherwise, E the left vector space $K_s^{(\mathbf{N})}$ and let $(e_n)_{n \in \mathbf{N}}$ be the canonical basis of this space. On the other hand let $\sigma$ be an endomorphism of K and $d$ a $(\sigma, 1_K)$-derivation (Exercise 1) of K into itself, in other words an additive mapping of K into itself such that

The elements of E are linear combinations of products $\alpha_n e_n$ and a multiplication is defined on E ($\mathbf{Z}$-bilinear mapping of $E \times E$ into E) by the conditions

Thus an (in general non-commutative) ring structure is defined on E with $e_0$ as unit element (identified with the element 1 of K). If we write $X = e_1$, then $e_n = X^n$ for $n \geq 1$, so that the elements of E can be written as with the multiplication rule This ring is denoted by $K[X; \sigma, d]$ and is called the ring of non-commutative polynomials in X with coefficients in K, relative to $\sigma$ and $d$. For a non-zero element $z = \sum_j \alpha_j X^j$ of E, the degree $\deg(z)$ is the greatest integer $j$ such that $\alpha_j \neq 0$.

(a) Show that if $u, v$ are two elements $\neq 0$ of E, then:

Moreover, if $\deg(u) \geq \deg(v)$, there exist two elements $w, r$ of E such that $u = wv + r$ and, either $r = 0$, or $\deg(r) < \deg(v)$ ("Euclidean division" in E).

(b) Conversely, let R be a ring such that there exists a mapping $u \mapsto \deg(u)$ of the set $R'$ of elements $\neq 0$ of R into $\mathbf{N}$, satisfying the conditions of (a). Show that the set K consisting of 0 and the non-zero elements $u \in R$ such that $\deg(u) = 0$ is a (not necessarily commutative) field. If $K \neq R$ and $x$ is an element of R for which the integer $\deg(x) > 0$ is the smallest possible, show that every element of R can be written uniquely in the form $\sum_i \alpha_i x^i$ with $\alpha_i \in K$. (To see that every element of R is of this form, argue by reductio ad absurdum by considering an element $u \in R$ which is not of this form and for which $\deg(u)$ is the smallest possible.) Prove that there exist an endomorphism $\sigma$ of K and a $(\sigma, l_K)$-derivation $d$ of K such that $x \alpha = \sigma(\alpha)x + d\alpha$ for all $\alpha \in K$. Deduce that R is isomorphic to a field or a ring $K[X; \sigma, d]$.