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Friday, February 19, 2021

Radical ideals in Artitian rings are boolean

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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