chore(Algebra/*/NonUnital*): split nonassoc and assoc nonunital center instances

[pechersky]

There is a diamond from how Subsemigroup.center defines mul and implies associativity and commutativity. So over a non-assoc parent type, the instance one gets differs than if you already had associativity. When/if we set up positive nat powers, there will be a diamond.

So at least prepare for that world by having the overriding instances in the "happy path" case.

---

---

Discovered while working on #40785.
Used GPT 5.6 Terra to classify hunks from the originating PR and do the cherry-pick operations.

[github-actions]

PR summary 1c1ec8b97e

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ center.instNonUnitalCommRingOfNonUnitalRing
+ instNonUnitalCommSemiringOfNonUnitalSemiring
+ instance (priority := 75) center.instNonUnitalCommRing : NonUnitalCommRing (center R)
+ instance (priority := 75) center.instNonUnitalCommSemiring :
+ instance (priority := 75) center.instNonUnitalCommSemiring : NonUnitalCommSemiring (center R)
+ instance (priority := 75) instNonUnitalCommSemiring : NonUnitalCommSemiring (center R₀ A₀)
++ center.instNonUnitalCommSemiringOfNonUnitalSemiring
- center.instNonUnitalCommRing
- instNonUnitalCommSemiring
-- center.instNonUnitalCommSemiring

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 1c1ec8b).

• +4 new declarations
• −0 removed declarations

+NonUnitalStarSubalgebra.instNonUnitalCommSemiringOfNonUnitalSemiring
+NonUnitalSubalgebra.center.instNonUnitalCommSemiringOfNonUnitalSemiring
+NonUnitalSubring.center.instNonUnitalCommRingOfNonUnitalRing
+NonUnitalSubsemiring.center.instNonUnitalCommSemiringOfNonUnitalSemiring

---

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

Current number	Change	Type (strong)
5616	-1	backward.isDefEq.respectTransparency

No changes to weak technical debt.

Current commit 1c1ec8b97e
Reference commit 1f165a51de

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).