Simple algebra

Remarks

Equivalently, the congruence lattice of $\mathbf{A}$ is the two-element chain. By the correspondence theorem, a quotient $\mathbf{A}/\theta$ is simple exactly when $\theta$ is a maximal element of $\operatorname{Con} \mathbf{A} \setminus \set{\nabla_A}$.

Example: the group $\Z/p\Z$ with $p$ prime is simple. Its congruences come from its subgroups, and by Lagrange's theorem these are only $\set{0}$ and $\Z/p\Z$.

Non-example: the ring $\Z$ is not simple. Congruence modulo $2$ is a congruence, and it is neither $\Delta_\Z$, since $(0, 2)$ belongs to it, nor $\nabla_\Z$, since $(0, 1)$ does not.

A field, viewed as a ring, is simple, since its only ideals are $\set{0}$ and the whole field. The one-element algebra is excluded, because there $\Delta_A = \nabla_A$.