Essential extension

Remarks

Equivalently, every homomorphism $\varphi$ from $B$ whose restriction to $A$ is injective is itself injective. An embedding $e \colon A \to B$ is called essential when $B$ is an essential extension of $e(A)$. For modules over a ring, congruences correspond to submodules, and the condition says that every nonzero submodule of $B$ meets $A$ in a nonzero element.

Example: $\Q$ is an essential extension of $\Z$ as abelian groups. Congruences of $\Q$ correspond to subgroups $N$, and $\theta \cap (\Z \times \Z) = \Delta_{\Z}$ means $N \cap \Z = \set{0}$. If $N$ contains $p/q \ne 0$, then it contains the nonzero integer $p$, so $N \cap \Z = \set{0}$ forces $N = \set{0}$.

Non-example: $\Z \times \Z$ is not an essential extension of $\Z \times \set{0}$ as abelian groups. The subgroup $N = \set{0} \times \Z$ meets $\Z \times \set{0}$ only in $0$, but $N \ne \set{0}$.