Extension and contraction of ideals

Remarks

The relations $I \subseteq I^{ec}$, $J^{ce} \subseteq J$, $I^{e} = I^{ece}$ and $J^{c} = J^{cec}$ hold. If $J$ is a prime ideal of $B$, then $J^c$ is a prime ideal of $A$, because $A/J^c$ embeds into the integral domain $B/J$. Example: for the inclusion $\Z \to \Z[i]$, $(2 + i)^c = 5\Z$, since $(2 + i) \cap \Z$ contains $5 = (2+i)(2-i)$ and is a proper ideal containing $5\Z$. Non-example of the analogous property for extension: $2\Z$ is prime in $\Z$, but $(2\Z)^e = 2\Z[i]$ is not prime, since $(1 + i)^2 = 2i$ lies in it while $1 + i$ does not. The image $f(I)$ itself need not be an ideal of $B$; for $\Z \subseteq \Q$, the image of $2\Z$ is not an ideal of $\Q$, and its extension is all of $\Q$.