[Merged by Bors] - feat(RingTheory/PicardGroup): invertible modules over semirings are locally free

[alreadydone]

Theorem 10.7 in Facets of module theory over semirings (Borger & Jun, https://arxiv.org/abs/2405.18645, Adv Math). Aristotle discovered a simpler proof using Invertible.bijective_of_surjective available in Mathlib. (The task started at 01:37 AM and Aristotle found the proof at 01:46 AM.)

Also generalizes lemmas and adds a new def about local semirings.

Co-authored-by: Aristotle (Harmonic) aristotle-harmonic@harmonic.fun

---

[github-actions]

PR summary 2346445d2d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ exists_finset_free_localization
+ exists_of_isUnit_sum
+ instance [IsLocalRing R] : Subsingleton (Pic R) := subsingleton_iffₛ.mpr fun M _ _ _ ↦ by
+ nonunitsAddSubmonoid
- instance (R) [CommRing R] [IsLocalRing R] : Subsingleton (Pic R)

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 2346445).

• +3 new declarations
• −0 removed declarations

+IsLocalRing.exists_of_isUnit_sum
+IsLocalRing.nonunitsAddSubmonoid
+Module.Invertible.exists_finset_free_localization

---

Decrease in strong tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (strong)
4514	-1	backward.defeqAttrib.useBackward

No changes to weak technical debt.

Current commit 2346445d2d
Reference commit 69b866d83a

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).

[riccardobrasca]

Thanks!

bors d+

[mathlib-bors]

✌️ alreadydone can now approve this pull request until 2026-07-24 08:58 UTC (in 2 weeks). To approve and merge, reply with bors r+. More detailed instructions are available here.

⚠️ This delegation only covers changes within Archive/**, Counterexamples/**, docs/**, DownstreamTest/**, Mathlib/**, MathlibTest/**, widget/**, Archive.lean, Counterexamples.lean, docs.lean, Mathlib.lean; an author commit touching anything else will revoke it. Bors also revokes it if a later push changes too many files for it to check the full list — even if it stays within scope.

[riccardobrasca]

@alreadydone Just take any monoid M with an element x that has a left inverse but no right inverse and consider the monoid algebra over M and coefficients in ℕ: this is local for trivial reasons, as for natural numbers a+b=0 implies a=b=0 and a+b=1 implies a=1, b=0 or a=0, b=1. Now the element corresponding to x (in degree 1) has a left inverse but no right inverse.

[riccardobrasca]

Wait, this is exactly essentially already written in mathoverlow! What am I missing?

[alreadydone]

Wait, this is exactly essentially already written in mathoverlow! What am I missing?

Indeed, I think I just forgot the example is a local semiring by mathlib definition, and thought the particular question was still not resolved. Thank you, and I'll update the MO question.

[riccardobrasca]

We can even add the instance IsLocalRing (MonoidAlgebra Nat M) to mathlib, it holds for any monoid!

[alreadydone]

Thanks 🎉
bors r+

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• ci (staging) / Test and lint
• ci (staging) / Build
• ci (staging) / Post-Build Step
• ci (staging) / Lint style
• ci (staging) / Post-CI job