Let $R$ be a commutative ring with unity. If for some $a,b,c\in R$, we have $x,y,z\in R$ such that $ax+by+cz=1$, then does it imply that $\exists x',y',z'\in R$ such that $a^{50}x'+b^{20}y'+c^{15}z'=1$.
By analogy with the ring of integers, I think this is true for a ring with unique factorization. Any hints? Thanks beforehand.
Notice $$ax+by+cz = 1\quad\implies\quad \underbrace{(ax+by+cz)^{83}}_{\rm LHS} = \underbrace{1}_{\rm RHS}$$ Expand LHS as a sum of monomials over $a,b,c$:
$${\rm LHS} = (ax+by+cz)^{83} = \sum_{p,q,r} \alpha_{pqr} a^pb^qc^r\quad\text{ with } \alpha_{pqr} \in R$$ For all these monomials, we have $p+q+r = 83$. We can formally rewrite LHS as
$$\begin{align}{\rm LHS} &= a^{50}\sum_{p \ge 50,q,r} \alpha_{pqr} a^{p-50}b^q c^r\\ &+ b^{20}\sum_{p < 50,q \ge 20,r} \alpha_{pqr} a^p b^{q-20} c^r\\ &+ c^{15}\sum_{p < 50, q < 20, r} \alpha_{pqr} a^p b^q c^{r-20}\end{align} $$
It is clear the coefficients for $a^{50}$ and $b^{20}$ belongs to $R$. Since $$83 > 82 = (50-1)+(20-1)+(15-1)$$ at least one of the following three conditions will be satisfied:
$$p \ge 50\quad\text{ or }\quad q \ge 20\quad\text{ or }\quad r \ge 15$$
In the coefficients of $c^{15}$, $p < 50$ and $q < 20$, this forces $r - 15 \ge 0$ and hence this coefficient also belong to $R$.
Define
$$\begin{align} x' &= \sum_{p \ge 50,q,r} \alpha_{pqr} a^{p-50}b^q c^r\\ y' &= \sum_{p < 50,q \ge 20,r} \alpha_{pqr} a^p b^{q-20} c^r\\ z' &= \sum_{p < 50, q < 20, r} \alpha_{pqr} a^p b^q c^{r-20}\end{align} $$ We have $x', y', z' \in R$ and $$a^{50}x' + b^{20} y' + c^{15}z' = {\rm LHS} = {\rm RHS} = 1$$