feat(RingTheory): definition of Gorenstein local ring

[Thmoas-Guan]

In this PR, we gave basic definition of Gorenstein local ring and Gorestein ring and prove that they are stable under ring equiv.

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• depends on: [Merged by Bors] - feat(Algebra/ModuleCat): injective dimension in linear equiv #41885

[github-actions]

PR summary e398b621ff

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.Gorenstein.Defs (new file)	2031

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Declarations diff (regex)

+ IsGorensteinLocalRing
+ IsGorensteinLocalRing.of_ringEquiv
+ IsGorensteinRing
+ isGorensteinLocalRing_def
+ isGorensteinRing_def
+ isGorensteinRing_def'
+ isGorensteinRing_of_ringEquiv

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit e398b62).

• +18 new declarations
• −0 removed declarations

+IsGorensteinLocalRing
+IsGorensteinLocalRing.casesOn
+IsGorensteinLocalRing.injdim
+IsGorensteinLocalRing.mk
+IsGorensteinLocalRing.of_ringEquiv
+IsGorensteinLocalRing.rec
+IsGorensteinLocalRing.recOn
+IsGorensteinLocalRing.toIsLocalRing
+IsGorensteinRing
+IsGorensteinRing.casesOn
+IsGorensteinRing.isGorenstein_localize
+IsGorensteinRing.mk
+IsGorensteinRing.rec
+IsGorensteinRing.recOn
+isGorensteinLocalRing_def
+isGorensteinRing_def
+isGorensteinRing_def'
+isGorensteinRing_of_ringEquiv

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5021	1	exposed public sections

Current commit e398b621ff
Reference commit 8f560a2211

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/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).

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - feat(Algebra/ModuleCat): injective dimension in linear equiv #41885
  By Dependent Issues (🤖). Happy coding!