[see also: need, entail]
For general rings, $ Out(R)$ is not necessarily well-behaved.
Clearly, $F$ is bounded but it is not necessarily so after division by $G$.
......where $P(d)$ denotes the space of (not necessarily monic) polynomial functions of degree $d$.
Necessarily, one of $X$ and $Y$ is in $Z$.
His proof is unnecessarily complicated.
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