Exercise 6, § 1

Let $K$ be a commutative field, $l$ a set and $L$ the free associative algebra on the set $l$ over the ring $K$ (III, p. 448, Def. 2) with the graduation defined in III, p. 458. We define the degree of an element of $L$ and the notion of homogeneous element as in IV, p. 2. a) Show that $\deg(uv) = \deg(u) + \deg(v)$ and $\deg(u + v) \leq \sup(\deg(u), \deg(v))$ for $u, v \in L$. b) Given $u, v, q, q' \in L$ such that $\deg(u) \geq \deg(v)$ and $\deg(qu - q'v) < \deg(qu)$; show that there exists $q'' \in L$ such that $\deg(u - q''v) < \deg(u)$. c) Given $u, v, q, q' \in L$ with $\deg u \geq \deg v > \deg(qu - q'v)$; show that there exists $q'' \in L$ such that $\deg(u - q''v) < \deg v$. d) Given $u, v, u', v' \in L$, homogeneous and such that $uv = u'v'$ and $\deg v \geq \deg v' \geq 0$; there exists $t \in L$ such that $u' = ut, v = tv'$. e) Given $u, v \in L$, homogeneous, such that $uv = vu$, there exists $w \in L, a, b \in K$ and $m, n \in K$ such that $u = aw^m, v = bw^n$. f) Let $u, v \in L$ be such that there exist $a, b \in L$, non-zero, with $au = bv$; there exist $m, d \in L$ such that the left multiples (resp. right divisors) common to $u$ and $v$ are left multiples (resp. right divisors) of $m$ (resp. $d$).