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 .

Version: v1 · 2026-09-28 · History
Domain-specific #: 11729
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Ring Theory, Operator Theory → Mathematics

Core Idea¶

Regular ideal is treated here as the recurring mathematics_logic_statistics 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 e in A such that ex - x \in \mathfrak{i} for every x \in A . In commutative algebra a regular ideal refers to an ideal containing a non-zero divisor.

A two-sided ideal \mathfrak{i} of a ring R can also be called a (von Neumann) regular ideal if for each element x of \mathfrak{i} there exists a y in \mathfrak{i} such that xyx=x. 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. Since the adjective regular has been overloaded, this article adopts the alternative adjectives modular, regular element, von Neumann regular, and quotient von Neumann regular to distinguish between concepts.

For Regular ideal, the abstraction is narrower than the article's general subject matter: a positive case must preserve 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. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — In a Marot ring, every regular ideal is generated by regular elements.
Constitutive relation — Since a is in J, so is rar, and so by setting y=rar we have the conclusion.
Operating condition — Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J.
Recognition evidence — The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.
Admissible variation — A two-sided ideal \mathfrak{i} is modular if and only if A/\mathfrak{i} is unital.
Characteristic consequence — In a unital ring, every ideal is modular since choosing e=1 works for any right ideal.
Failure boundary — So, the notion is more interesting for non-unital rings such as Banach algebras.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by 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 .
Not an over-broad reading. However, it is possible for a ring without identity to lack modular right ideals entirely.
Not an over-broad reading. This gives an example of an ideal which is not a regular element ideal.
Not an over-broad reading. Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J.
Not automatically Primal ideal. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Regular ideal applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

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.
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.
Properties and examplesModular ideals. A two-sided ideal \mathfrak{i} is modular if and only if A/\mathfrak{i} is unital.
Properties and examplesModular ideals. In a unital ring, every ideal is modular since choosing e=1 works for any right ideal.
Properties and examplesModular ideals. So, the notion is more interesting for non-unital rings such as Banach algebras.
Properties and examplesModular ideals. From the definition it is easy to see that an ideal containing a modular ideal is itself modular.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

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 . The strongest recognition evidence in the frozen account is: The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However, it is possible for a ring without identity to lack modular right ideals entirely. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Regular ideal compresses multiple mathematics_logic_statistics 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. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics 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.
Demand recognition evidence. The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.
Test variation. Change an implementation or setting while preserving a two-sided ideal \mathfrak{i} is modular if and only if A/\mathfrak{i} is unital.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

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. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

So, the notion is more interesting for non-unital rings such as Banach algebras. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → 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 ; recognition evidence → The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings

Applied / In Practice¶

The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Properties and examplesModular ideals; invariant → 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 ; boundary → the case exits the class when however, it is possible for a ring without identity to lack modular right ideals entirely

Structural Tensions¶

T1 — Stable identity versus admissible variation. However, it is possible for a ring without identity to lack modular right ideals entirely. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. This gives an example of an ideal which is not a regular element ideal. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. A simple domain which is not a division ring has the minimum possible number of von Neumann regular ideals: only the {0} ideal. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. In a Marot ring, every regular ideal is generated by regular elements. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Regular ideal literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. Since a is in J, so is rar, and so by setting y=rar we have the conclusion. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Regular ideal distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Regular ideal is structural-leaning. Its structural side is the repeatable organization summarized by 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 . Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. 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 . The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: In a Marot ring, every regular ideal is generated by regular elements. Since a is in J, so is rar, and so by setting y=rar we have the conclusion. It further constrains recognition and variation through: Let R Be a local ring which is not a division ring, and denote the unique maximal right ideal by J. The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings.

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Regular ideal literal. Its documented scope includes the condition that 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. Another bounded application condition is that The notion of modular ideals permits the generalization of various characterizations of ideals in a unital ring to non-unital settings. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—A two-sided ideal \mathfrak{i} is modular if and only if A/\mathfrak{i} is unital.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Ring Ideal.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Regular ideal. The reviewed identity 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. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Regular ideal Domain-specific

Parents (1) — more general patterns this builds on

Regular ideal is a kind of Ring Ideal Domain-specific

It is a right ideal satisfying an additional modularity condition.

Neighborhood in Abstraction Space¶

Regular ideal sits in a moderately populated region (40^th 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

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish 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 ?
Primal ideal. Primal ideal denotes proper ideal of a commutative ring such that the elements not prime to it form an ideal in ring theory. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Principal Ideal. Form the smallest ideal containing one ring element, using all allowed left, right, or two-sided ring multiples and additive closure. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Primitive ideal. Annihilator of a simple module. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Regular ideal remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Regular_ideal (revision 1348074780).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.