Extension of an ideal ($I^e$)

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.