Given a commutative ring \(R\), an ideal \(I \subset R\) is a subset such that \(f g \in I\) whenever \(f \in R\) and \(g \in I\) and \(g+h \in I\) whenever \(g,h \in I\). Show that if \(I \subsetneq ...Given a commutative ring \(R\), an ideal \(I \subset R\) is a subset such that \(f g \in I\) whenever \(f \in R\) and \(g \in I\) and \(g+h \in I\) whenever \(g,h \in I\). Show that if \(I \subsetneq \mathcal{O}_p\) is a proper ideal (ideal such that \(I \not= \mathcal{O}_0\)), then \(I \subset \mathfrak{m}_p\), that is \(\mathfrak{m}_p\) is a maximal ideal.
