Corollary 2

For an ideal $a$ of a ring $A$ we denote by $r(a)$ the radical of $a$. Then, for two ideals $a, b$ of $A$,

in particular, if $a \subset b$, then $r(a) \subset r(b)$.

For a prime ideal to contain $a \cap b$ (or $ab$), it is necessary and sufficient that it contain one of the ideals $a, b$ (§ 1, no. 1, Proposition 1).