The lattice of ideals of a commutative ring $R$ form a modular lattice as does the sublattice of intermediate normal field extensions of a Galois extension. The lattice of radical ideals forms a distributive lattice as does the lattice of subfields of a finite field. Finally, the lattice of radical ideals of Artinian rings form a boolean algebra. And those are only the subalgebra lattices associated to commutative rings.
Corollary. $Spec(R)$ in an Artinian ring forms a finite discrete topology
Let $R$ be an Artinian ring, then $R$ is krull dimension zero, which means all prime ideals are maximal [1]. Additionally, there are finitely many such maximal ideals [1]. The first means that $Spec(R)$ is T1 and the second means that it is finite. All finite T1 spaces are discrete. $\square$
Corollary. the lattice of radical ideals of an Artitian ring forms a boolean algebra
The closed sets of $Spec(R)$ are a finite discrete topology, which is a power set. By basic set theory, the power sets of sets form a complete atomic boolean algebra. $Spec(R)$ is order dual to the lattice of radical ideals, so that means that the lattice of radical ideals of an Artinian ring also forms a boolean algebra. $\square$
A final comment is worthwhile on the subject of direct product decompositions. Artinian rings are the direct product of local Artinian rings [1], which only have a single maximal ideal. Finite boolean algebras on the other hand are direct products of ordered pairs. The two types of structure therefore are direct products.
Source:
[1] Zariski commutative algebra volume one part four
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