[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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