Residually finite algebra

Remarks

Equivalently, $A$ is isomorphic to a subdirect product of finite algebras. Equivalently, the congruences of $A$ with finite quotient intersect to the trivial congruence. For groups, this is the definition of a residually finite group.

Example: the group $(\Z, +)$ is residually finite. For $a \ne b$, let $n = \abs{a - b} + 1$; the reduction map $\Z \to \Z/n\Z$ sends $a$ and $b$ to different classes, since $n$ does not divide $a - b$.

Non-example: the group $(\Q, +)$ is not residually finite. A homomorphic image of $\Q$ is divisible, and a finite divisible abelian group is trivial, since multiplication by its order is both zero and onto. So every homomorphism from $\Q$ to a finite group sends $0$ and $1$ to the same element.