feat(RingTheory/Artinian/Ring): an Artinian ring is isomorphic to the product of its localizations

[tb65536]

This PR proves that an Artinian ring is isomorphic to the product of its localizations.

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The best proof I could come up with goes via the quotient by the power of the nilradical, but it's rather clunky, so alternate proofs more than welcome!

[github-actions]

PR summary 044631201f

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.Artinian.Ring	1369	1374	+5 (+0.37%)
Mathlib.RingTheory.Localization.AtPrime.Basic	1222	1223	+1 (+0.08%)

Import changes for all files

Files	Import difference
12 files Mathlib.RingTheory.Ideal.MinimalPrime.Localization Mathlib.RingTheory.LocalRing.LocalSubring Mathlib.RingTheory.Localization.AsSubring Mathlib.RingTheory.Localization.AtPrime.Basic Mathlib.RingTheory.Localization.LocalizationLocalization Mathlib.RingTheory.QuasiFinite.Basic Mathlib.RingTheory.QuasiFinite.Polynomial Mathlib.RingTheory.QuasiFinite.Weakly Mathlib.RingTheory.RingHom.QuasiFinite Mathlib.RingTheory.SurjectiveOnStalks Mathlib.RingTheory.Trace.Quotient Mathlib.Topology.Algebra.Ring.Compact	1
Mathlib.RingTheory.Artinian.Ring Mathlib.RingTheory.LocalRing.Quotient	5

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Declarations diff

+ equivPiLocalizationAux
+ equivPiLocalizationMaximal
+ equivPiLocalizationMaximal_apply
+ equivPiLocalizationMaximal_apply_apply
+ equivPiLocalizationPrime
+ equivPiLocalizationPrime_apply
+ equivPiLocalizationPrime_apply_apply
+ injective_algebraMap_pi_localization_maximalSpectrum

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).