Regular ideal¶
In operator theory, a right ideal \mathfrak{i} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that ex - x \in \mathfrak{i} for every x \in A .
Core Idea¶
Regular ideal is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In operator theory, a right ideal \mathfrak{i} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that ex - x \in \mathfrak{i} for every x \in A . In mathematics, especially ring theory, a regular ideal can refer to multiple concepts. In operator theory, a right ideal \mathfrak{i} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element.
Scope of Application¶
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Documented setting. Finally, regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R/J is von Neumann regular ring.
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Properties and examplesModular ideals. The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.
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Properties and examplesModular ideals. A two-sided ideal \mathfrak{i} is modular if and only if A/\mathfrak{i} is unital.
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Properties and examplesModular ideals. In a unital ring, every ideal is modular since choosing e=1 works for any right ideal.
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Properties and examplesModular ideals. So, the notion is more interesting for non-unital rings such as Banach algebras.
Clarity¶
A clear use of Regular ideal names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In operator theory, a right ideal \mathfrak{i} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that ex - x \in \mathfrak{i} for every x \in A .
Manages Complexity¶
Regular ideal compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—since a is in J, so is rar, and so by setting y=rar we have the conclusion.—and the practical consequence—in a unital ring, every ideal is modular since choosing e=1 works for any right ideal.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In operator theory, a right ideal \mathfrak{i} in a (possibly) non-unital ring A is said to be regular (or modular) if there exists an element e in A such that ex - x \in \mathfrak{i} for every x \in A .
- Check operation and conditions. Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J.
Knowledge Transfer¶
Within the home domain. Knowledge about Regular ideal transfers literally when a new case preserves the same carrier type, relation, and recognition test. Finally, regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R/J is von Neumann regular ring. The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings. Beyond the home domain. No canonical parent is asserted for Regular ideal.
Relationships to Other Abstractions¶
Current abstraction Regular ideal Domain-specific
Parents (1) — more general patterns this builds on
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Regular ideal is a kind of Ring Ideal Domain-specific
It is a right ideal satisfying an additional modularity condition.
Hierarchy paths (6) — routes to 5 parentless roots
- Regular ideal → Ring Ideal → Set and Membership
- Regular ideal → Ring Ideal → Ring → Group → Monoid → Identity Element
- Regular ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Closure
- Regular ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Set and Membership
- Regular ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Associativity → Invariance
- Regular ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Regular ideal sits in a moderately populated region (40th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Ring & Module Structure Theory (14 abstractions)
Nearest neighbors
- Idealizer — 0.89
- Rees decomposition — 0.87
- Dualizing module — 0.87
- Group Ring — 0.87
- Length of a module — 0.87
Computed from structural-signature embeddings · 2026-10-08