Take the 2-minute tour ×  Given a unital ring $R$ and its unital
subring $P$ (with the same unit). Also, given a maximal left
ideal $L$ of $R$. Must $L\cap P$ be a maximal left ideal of $P$?
I don't think so, but are there any conditions on $R$ or $P$
which ensure that?  share|improve this question add comment  2
Answers  up vote 2 down vote accepted  No, $L\cap P$ is not
necessarily a maximal left ideal of $P$.  Example:If
$R=\mathbb{R}$, $P=\mathbb{Z}$, then $L=(0)$ is a maximal ideal
of $R$ because $R$ is a field. So $L\cap P=(0)$ but $(0)$ is not
a maximal ideal in $\mathbb{Z}$, because $(0) \subset
2\mathbb{Z} \subset \mathbb{Z}$  share|improve this answer add
comment  The answer in general is no (see francis-jamet's
answer) but there is at least one important case where the
answer is affirmative.  If $S\subset R$ is an integral extension
and $\frak m$ is a maximal ideal in $R$ then ${\frak m}\cap S$
is a maximal ideal in $S$.  This situation is very relevant for
instance in algebraic number theory where