Let $R$ be a commutative ring and $P$ be a prime ideal.
Consider the quotient Ring $R/P$ and localization $R_P$, in the Borcherd's lecture, he said $R/P$ making $P$ minimal, but $R_P$ making $P$ maximal.
Can someone explain this to me in more detail?
In $R/P$, $P$ has been shrunk to a single element (the coset $0+P$) so $P/P=\{0+P\}$ is contained in all proper ideals of the quotient and can't get any smaller.
On the other hand, in $R_P$, things outside of $P$ have been converted to units, so the image of $P$ in the localization is the unique maximal ideal, containing all other proper ideals of the localization.