Primitive ideal¶
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module.
Core Idea¶
Primitive ideal is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. The quotient of a ring by a left primitive ideal is a left primitive ring.
Scope of Application¶
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Primitive spectrum. The primitive spectrum of a ring is a non-commutative analog of the prime spectrum of a commutative ring.
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Primitive spectrum. Let A be a ring and \operatorname{Prim}(A) the set of all primitive ideals of A.
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Primitive spectrum. Then there is a topology on \operatorname{Prim}(A) , called the Jacobson topology, defined so that the closure of a subset T is the set of primitive ideals of A containing.
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Primitive spectrum. Then, by definition, a primitive ideal is the kernel of an irreducible representation \pi of A and thus there is a surjection.
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Primitive spectrum. Now, suppose A is an associative algebra over a field.
Clarity¶
A clear use of Primitive ideal names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module.
Manages Complexity¶
Primitive ideal compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—the quotient of a ring by a left primitive ideal is a left primitive ring.—and the practical consequence—now, suppose A is an associative algebra over a field. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module.
- Check operation and conditions. The primitive spectrum of a ring is a non-commutative analog of the prime spectrum of a commutative ring.
- Demand recognition evidence. Let A be a ring and \operatorname{Prim}(A) the set of all primitive ideals of A.
- Test variation.
Knowledge Transfer¶
Within the home domain. Knowledge about Primitive ideal transfers literally when a new case preserves the same carrier type, relation, and recognition test. The primitive spectrum of a ring is a non-commutative analog of the prime spectrum of a commutative ring. Let A be a ring and \operatorname{Prim}(A) the set of all primitive ideals of A. Beyond the home domain. No canonical parent is asserted for Primitive 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.
Relationships to Other Abstractions¶
Current abstraction Primitive ideal Domain-specific
Parents (1) — more general patterns this builds on
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Primitive ideal is a kind of Ring Ideal Domain-specific
It is an annihilator ideal of a simple module.
Hierarchy paths (6) — routes to 5 parentless roots
- Primitive ideal → Ring Ideal → Set and Membership
- Primitive ideal → Ring Ideal → Ring → Group → Monoid → Identity Element
- Primitive ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Closure
- Primitive ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Set and Membership
- Primitive ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Associativity → Invariance
- Primitive ideal → Ring Ideal → Ring → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Primitive ideal sits in a moderately populated region (49th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Spectrum of a C*-Algebra — 0.88
- Primal ideal — 0.87
- Semiprimitive ring — 0.86
- Nilradical of a ring — 0.85
- Regular ideal — 0.85
Computed from structural-signature embeddings · 2026-10-08