Jaffard ring¶
In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions.
Core Idea¶
Jaffard ring is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960.
Scope of Application¶
-
Documented setting. In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions.
-
Documented setting. They are named for Paul Jaffard who first studied them in 1960.
-
Documented setting. Formally, a Jaffard ring is a ring R such that the polynomial ring.
-
Documented setting. A Jaffard ring that is also an integral domain is called a Jaffard domain.
-
Documented setting. The Jaffard property is satisfied by any Noetherian ring R, and examples of non-Noetherian rings might appear to be quite difficult to find, however they do arise naturally.
Clarity¶
A clear use of Jaffard ring names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions.
Manages Complexity¶
Jaffard ring compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—another example is obtained by "pinching" formal power series at the origin along a subfield of infinite extension degree, such as the subring of \overline{\mathbf{Q}} T consisting of those formal power series whose constant term is rational.—and the practical consequence—a Jaffard ring that is also an integral domain.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions.
- Check operation and conditions. In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions.
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Jaffard ring transfers literally when a new case preserves the same carrier type, relation, and recognition test. In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960. Beyond the home domain. No canonical parent is asserted for Jaffard ring.
Relationships to Other Abstractions¶
Current abstraction Jaffard ring Domain-specific
Parents (1) — more general patterns this builds on
-
Jaffard ring is a kind of Ring Domain-specific
A Jaffard ring is a ring distinguished by controlled Krull-dimension behavior under polynomial extension.
Hierarchy paths (5) — routes to 5 parentless roots
- Jaffard ring → Ring → Group → Monoid → Semigroup → Set and Membership
Neighborhood in Abstraction Space¶
Jaffard ring sits in a sparse region of the domain-specific corpus (69th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Ring & Module Structure Theory (14 abstractions)
Nearest neighbors
- Regular ideal — 0.85
- Finite extensions of local fields — 0.84
- Group Ring — 0.84
- Zero Divisor — 0.84
- Dualizing module — 0.83
Computed from structural-signature embeddings · 2026-10-08