Rees decomposition¶
In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings.
Core Idea¶
Rees decomposition is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings. In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings. where each η α is a homogeneous element and the d elements θ i are a homogeneous system of parameters for R and. Suppose that a ring R is a quotient of a polynomial ring k[x 1 ,...] over a field by some homogeneous.
Scope of Application¶
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Definition. Suppose that a ring R is a quotient of a polynomial ring k[x 1 ,...] over a field by some homogeneous ideal.
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Definition. A Rees decomposition of R is a representation of R as a direct sum (of vector spaces).
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Definition. where each η α is a homogeneous element and the d elements θ i are a homogeneous system of parameters for R and.
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Definition. η α k[θ f α +1 ,...,θ d ] ⊆ k[θ 1 , θ f α ].
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Definition. R = \bigoplus\alpha \eta\alpha k[\theta1,\ldots,\theta{f\alpha}].
Clarity¶
A clear use of Rees decomposition 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 Rees decomposition is a way of writing a ring in terms of polynomial subrings. The strongest recognition evidence in the frozen account is: η α k[θ f α +1 ,...,θ d ] ⊆ k[θ 1 , θ f α ].
Manages Complexity¶
Rees decomposition compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—a Rees decomposition of R is a representation of R as a direct sum (of vector spaces).—and the practical consequence—in commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings.
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 Rees decomposition is a way of writing a ring in terms of polynomial subrings.
- Check operation and conditions. where each η α is a homogeneous element and the d elements θ i are a homogeneous system of parameters for R and.
- Demand recognition evidence. η α k[θ f α +1 ,...,θ d ] ⊆ k[θ 1 , θ f α ]. 5.
Knowledge Transfer¶
Within the home domain. Knowledge about Rees decomposition transfers literally when a new case preserves the same carrier type, relation, and recognition test. Suppose that a ring R is a quotient of a polynomial ring k[x 1 ,...] over a field by some homogeneous ideal. A Rees decomposition of R is a representation of R as a direct sum (of vector spaces). Beyond the home domain. Transfer the broader Decomposition relation when the mathematics logic statistics-specific differentia cannot be filled. Retain the name Rees decomposition only when the same carrier, operation, and rejection conditions are present literally rather than metaphorically.
Relationships to Other Abstractions¶
Current abstraction Rees decomposition Domain-specific
Parents (1) — more general patterns this builds on
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Rees decomposition is a kind of Decomposition Prime
Rees decomposition is a strict kind of Decomposition: In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings.
Hierarchy path (1) — routes to 1 parentless root
- Rees decomposition → Decomposition
Neighborhood in Abstraction Space¶
Rees decomposition sits in a crowded region of the domain-specific corpus (33rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Polynomials & Algebraic Invariants (20 abstractions)
Nearest neighbors
- Laurent Polynomial — 0.89
- Group Ring — 0.88
- Length of a module — 0.88
- Supermodule — 0.88
- Idealizer — 0.88
Computed from structural-signature embeddings · 2026-10-08