Marot ring¶
A commutative ring in which every ideal containing a non-zero-divisor is generated by the non-zero-divisors it contains.
Core Idea¶
A Marot ring is a commutative ring with identity whose regular ideals are generated by the regular elements they contain.[1] An element is regular here when it is a non-zero-divisor; an ideal is regular when it contains at least one such element.[1] The requirement concerns every qualifying ideal in the ring. One ideal with a regular generator is an illustration, not a proof that the whole ring is Marot.
The point of the rule is that an ideal may contain zero divisors, yet its regular members suffice to generate the entire ideal. That is an additional condition on a commutative ring; it does not follow merely from the ring operations.
Scope of Application¶
The property belongs to commutative algebra, especially ideal theory in rings that may have zero divisors. A mathematician can test a ring by asking whether each ideal that contains a non-zero-divisor can be generated from the non-zero-divisors already in that ideal. Finite products of domains and Noetherian rings are two classes known to satisfy the property.[1]
Clarity¶
“Regular ideal” is easily misread. It does not mean that all its elements are regular; it only has to contain one regular element. Nor does it mean the modular right-ideal condition that appears under the same name elsewhere in ring theory. The Marot question asks whether the ideal's own regular elements generate all its members.
Manages Complexity¶
Instead of choosing a separate mixture of regular and zero-divisor generators for each qualifying ideal, the Marot property identifies one permitted pool: the ideal's regular elements. A theorem covering a whole ring class can establish this for every qualifying ideal at once. A calculation for one ideal helps explain the rule but cannot replace the ring-wide proof.
Abstract Reasoning¶
First identify the non-zero-divisors of the proposed ring. For any ideal containing one of them, compare the entire ideal with the ideal generated by its regular members. A counterexample among these qualifying ideals defeats the Marot claim. Alternatively, prove that the ring belongs to a class already known to be Marot, such as a finite product of domains or a Noetherian ring.[1]
Knowledge Transfer¶
The same test can be applied to differently constructed commutative rings because “non-zero-divisor,” “ideal,” and “generation” retain their literal algebraic meanings. It does not turn into a general claim that every system can be built from its “regular” parts; outside ring theory those words do not supply the required test.
Example¶
In \(\mathbb Z\times\mathbb Z\), multiplication is coordinatewise. A pair is a non-zero-divisor exactly when both coordinates are nonzero. The ideal \(2\mathbb Z\times3\mathbb Z\) is generated by the regular element \((2,3)\), since its products with arbitrary pairs \((m,n)\) are exactly \((2m,3n)\). This is a direct calculation illustrating one qualifying ideal. The separate published result that finite products of domains are Marot establishes the full ring-wide claim.[1]
Relationships to Other Abstractions¶
Current abstraction Marot ring Domain-specific
Parents (1) — outgoing typed relations
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Marot ring is a kind of Commutative ring Domain-specific
A Marot ring is a commutative ring with the additional rule that every regular ideal is generated by its regular elements.
Hierarchy paths (2) — routes to 2 parentless roots
- Marot ring → is a kind of Commutative ring → is a kind of Commutativity → is a kind of Invariance
- Marot ring → is a kind of Commutative ring → is a kind of Commutativity → is a kind of Symmetry
Neighborhood in Abstraction Space¶
Marot ring sits in a moderately populated region (45th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Ring & Module Structure Theory (15 abstractions)
Nearest neighbors
- Ring Ideal — 0.88
- Regular ideal — 0.87
- Idealizer — 0.86
- Ring — 0.86
- Difference of Two Squares — 0.86
Computed from structural-signature embeddings · 2026-10-25
Not to Be Confused With¶
- Commutative ring: has commutative multiplication but need not satisfy the extra Marot rule on regular ideals.
- Ring ideal: is an individual subobject; a Marot ring is a class of whole rings classified through all qualifying ideals.
- Regular ideal in the modular sense: is a different condition on an individual right ideal, including possibly nonunital rings.
- v-Marot ring: applies a related divisorial-ideal condition. Ordinary Marot implies v-Marot in the cited paper, but the definitions must be kept distinct.[1]
- Noetherian ring: is always Marot by the cited proposition, yet Noetherianity is a sufficient condition rather than the definition.[1]
References¶
[1] Alfred Geroldinger, Sebastian Ramacher, and Andreas Reinhart, “On v-Marot Mori Rings and C-Rings,” Journal of the Korean Mathematical Society 52, no. 1 (2015): 1–21, doi:10.4134/JKMS.2015.52.1.001. Author-hosted full journal reprint inspected: printed pp. 3–4 for conventions, definition, and product class; p. 7 Proposition 3.3(1) for the Noetherian and v-Marot implications. The particular product-ideal calculation above is a deduction, not an example printed in the paper. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g