= Extension of an ideal
{title2=$I^e$}
For a <ring homomorphism> $f:R\to A$, the extension $I^e$ of a two-sided <ideal> $I\subseteq R$ is the smallest <two-sided ideal> of $A$ containing $f(I)$. Thus its elements are finite sums $\sum_j a_j f(i_j)b_j$ with $i_j\in I$ and $a_j,b_j\in A$.
When $A$ is commutative, the formula becomes $I^e=IA=Af(I)$. For example, extending $(2)$ from $\mathbb Z$ to $\mathbb Z[i]$ gives the multiples of two in the Gaussian <integers>; extending it to $\mathbb Q$ gives the whole <field> because two is invertible there. Extension can therefore enlarge an <ideal> substantially even when the original <ring> map is injective.
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