Acceptable ring¶

In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7842
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Commutative Algebra → Mathematics

Core Idea¶

Acceptable ring is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings. In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.

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The Swapped-Rule Club

Mathematicians study 'number worlds' where you can add and multiply. They have a club for extra well-behaved number worlds, and to get in you must pass some tests. An acceptable ring is a member of a bigger club where some of those tests are swapped for gentler ones, so more number worlds can get in.

Excellent's Bigger Cousin

In math, a ring is a set of things you can add, subtract and multiply, like the whole numbers. Mathematicians have a list of conditions that make a ring 'excellent,' which means it behaves nicely in certain ways, and some of those conditions talk about a type of ring called 'regular.' An acceptable ring uses the same kind of list, but swaps the conditions about regular rings for conditions about a looser type called 'Gorenstein' rings. So acceptable rings are a bigger, more general group than excellent ones.

Gorenstein Version of Excellence

In commutative algebra, excellent rings are rings defined by a list of conditions that guarantee good behavior, and part of that definition refers to regular rings. An acceptable ring generalizes this: it keeps the same shape of definition but replaces the conditions about regular rings with conditions about Gorenstein rings, a broader class. Several families are known to be acceptable: every finite-dimensional Gorenstein ring, every finitely generated algebra over an acceptable ring, and every localization of an acceptable ring. The last two facts mean that standard ways of building new rings from old ones keep you inside the class.

An acceptable ring is a commutative-algebra notion obtained by modifying the definition of an excellent ring: the conditions in that definition that refer to regular rings are replaced by corresponding conditions referring to Gorenstein rings. Since the Gorenstein property is weaker than regularity, the resulting class generalizes excellent rings. The class has useful stability properties: all finite-dimensional Gorenstein rings are acceptable, finitely generated algebras over an acceptable ring are acceptable, and localizations of acceptable rings are acceptable. Closure under finitely generated algebras and localization is what makes the notion workable, because those are the main constructions used in practice. A positive instance must preserve the specific move of substituting Gorenstein conditions for regularity conditions within the excellence template, not merely a ring that behaves well in some informal sense.

Scope of Application¶

Documented setting. In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.
Documented setting. All finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.
Documented setting. In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.
Documented setting. All finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.
Documented setting. In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.

Clarity¶

A clear use of Acceptable ring names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.

Manages Complexity¶

Acceptable ring compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—all finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.—and the practical consequence—all finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings.
Check operation and conditions.

Knowledge Transfer¶

Within the home domain. Knowledge about Acceptable ring transfers literally when a new case preserves the same carrier type, relation, and recognition test. In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings. All finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings. Beyond the home domain. No canonical parent is asserted for Acceptable ring.

Relationships to Other Abstractions¶

Current abstraction Acceptable ring Domain-specific

Parents (1) — more general patterns this builds on

Acceptable ring is a kind of Ring Domain-specific

An acceptable ring is a ring whose excellence-style regularity conditions are replaced by Gorenstein conditions.

Neighborhood in Abstraction Space¶

Acceptable ring sits in a moderately populated region (59^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Algebraic Structures & Order Relations (18 abstractions)

Nearest neighbors

Primal ideal — 0.86
Regular ideal — 0.85
Zero Divisor — 0.85
Group Ring — 0.85
Idealizer — 0.85

Computed from structural-signature embeddings · 2026-10-08