Exercise 6, § 1

Soient $K$ a commutative field, $I$ a set, and $L$ the free associative algebra of the set $I$ over the ring $K$ (III, p. 21, déf. 2), provided with the grading defined in III, p. 31. Define the degree of an element of $L$, and the notion of homogeneous element, as in IV, p. 2. a) Prove that $\deg(uv) = \deg(u) + \deg(v)$ and $\deg(u + v) \leqslant \sup(\deg(u), \deg(v))$ for $u, v \in L$. b) Let $u, v, q, q' \in L$, such that $\deg(u) \geqslant \deg(v)$ and $\deg(qu - q'v) < \deg(qu)$; prove that there exists $q'' \in L$ such that $\deg(u - q''v) < \deg(u)$. c) Let $u, v, q, q' \in L$ with $\deg(u) \geqslant \deg(v) > \deg(qu - q'v)$; prove that there exists $q'' \in L$ such that $\deg(u - q''v) < \deg(v)$. d) Let $u, v, u', v' \in L$, homogeneous, and such that $uv = u'v'$ and $\deg(v) \geqslant \deg(v') \geqslant 0$; there exists $t \in L$ such that $u' = ut, v = tv'$. e) Let $u, v \in L$, homogeneous, such that $uv = vu$. There exists $w \in L, a, b \in K$, and $m, n \in \mathbf{N}$, such that $u = aw^m, v = bw^n$. f) Let $u, v \in L$ such that there exist $a, b \in L$, nonzero, with $au = bv$; there exist $m, d \in L$ such that the common left multiples (resp. right divisors) of $u$ and $v$ are the left multiples (resp. right divisors) of $m$ (resp. $d$).