Marot ring¶

A commutative ring in which every ideal containing a non-zero-divisor is generated by the non-zero-divisors it contains.

Version: v1 · 2026-10-25 · History
Domain-specific #: 14415
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Commutative Algebra, Ideal Theory → Mathematics

Core Idea¶

A Marot ring is a commutative ring with identity in which every regular ideal is generated by its regular elements.^[1] Here a regular element is a non-zero-divisor, and a regular ideal is an ideal that contains at least one such element.^[1] The quantifiers matter: for every ideal \(I\) containing a non-zero-divisor, the elements of \(I\) that are themselves non-zero-divisors must generate all of \(I\). A convenient symbolic statement is \(I=\langle I\cap R^{\bullet}\rangle\) for every regular ideal \(I\), where \(R^{\bullet}\) denotes the regular elements of \(R\). This is the defining equality, not a theorem that holds in all commutative rings.

The condition concerns generation of ideals, rather than merely whether a ring has zero divisors. It is useful precisely when a regular ideal may also contain zero divisors: it asks whether those members can be recovered from regular members of the same ideal. Showing that one selected ideal has a regular generator illustrates the test but does not establish that an entire ring is Marot. The cited paper gives two sufficient classes used below: finite products of domains and Noetherian rings.^[1]

Structural Signature¶

Sig role-phrases: commutative unital ring; regular-element filter; qualifying ideal; universal generation equality.

Commutative unital ring as bearer. The ring supplies addition, multiplication, and the ideals being classified. The cited work explicitly uses commutative rings with identity; changing that convention would require a new definition audit.^[1]
Regular-element filter. Multiplication in the bearer identifies non-zero-divisors. These elements are the permitted generators in the Marot condition. Allowing every ideal member as a generator would make the equality automatic and erase the differentia.
Qualifying ideal. An ideal enters the test only when it contains at least one regular element. An ideal made entirely of zero divisors is outside the quantified class, rather than a counterexample for lacking regular generators.^[1]
Universal generation equality. For each qualifying ideal \(I\), the ideal generated by its regular members must equal \(I\). A failure in one regular ideal defeats the ring-wide classification; a success in one regular ideal cannot prove it.^[1]

What It Is Not¶

Not a property of one ideal. “This ideal has a non-zero-divisor generator” is a local fact; Marot is a universal property of the ring's regular ideals.
Not the claim that every ideal has a regular generator. The condition applies only to ideals that already contain a non-zero-divisor, and it may require many regular generators.
Not the modular right-ideal sense of “regular ideal.” The live catalog's Regular ideal entry takes the modular right-ideal condition in a possibly nonunital ring as its main identity. In this entry “regular ideal” has the commutative-algebra meaning: it contains a non-zero-divisor.
Not v-Marot by substitution. The cited paper treats v-Marot as a related condition on regular divisorial ideals and proves ordinary Marot implies v-Marot.^[1] That direction does not license identifying the two definitions.
Not a synonym for Noetherian. Every Noetherian ring is Marot according to the cited proposition, but Noetherianity is a sufficient class, not the defining test.^[1]

Scope of Application¶

The literal scope is commutative algebra with the unital convention of the cited paper.^[1] The question arises when a proof or construction uses regular ideals in a ring that may have zero divisors and needs to know whether regular elements alone recover those ideals. The condition can be checked directly ideal by ideal or obtained from a sufficient class theorem. In the latter route, one must state why the actual carrier belongs to that class.

Finite products of domains provide one habitat: factors without zero divisors combine into a ring that has coordinatewise zero divisors but still satisfies the Marot condition.^[1] The paper also proves that a product of two rings is Marot exactly when both factors are.^[1] Noetherian rings provide a different habitat through Proposition 3.3(1).^[1] That proposition applies to a Noetherian quotient such as \(k[x,y]/(xy)\) once its Noetherian status is established; the named quotient is our mathematical instance of the published class theorem, not a worked example named in the paper.

Clarity¶

The adjective regular has two jobs that should be kept distinct. Applied to an element, it means non-zero-divisor.^[1] Applied to an ideal here, it means the ideal contains at least one non-zero-divisor; it does not say every element of the ideal is regular.^[1] The Marot condition then asks whether the subset of regular elements already in that ideal generates the entire ideal.^[1]

This distinction prevents two false shortcuts. First, an ideal with zero divisors among its members can still pass if those members are generated by its regular members. Second, observing that an ideal contains one regular element does not imply that this element, or even all regular elements in the ideal, generate it. The generation equality must be proved or supplied by a theorem that covers the whole ring.

Manages Complexity¶

An arbitrary ideal can have many generators and many zero-divisor members. The Marot property compresses the relevant question to a single admissible generating pool, \(I\cap R^{\bullet}\), for each regular ideal \(I\). Once the property is established for a ring, one can describe any of its regular ideals using only regular members without separately constructing a mixed generating set for each one. The compression is conditional: the definition itself does not tell us that an untested ring has the property.

Class results save a second layer of repeated work. For a finite product of domains, the source supplies the ring-wide conclusion.^[1] For a Noetherian ring, Proposition 3.3(1) does the same.^[1] A worked ideal remains valuable as a transparent illustration of the roles, but it should not be mistaken for the class proof.

Abstract Reasoning¶

Start by fixing the ring and its convention. Determine which elements are non-zero-divisors, then restrict attention to ideals containing at least one of them. For a proposed regular ideal \(I\), compare \(I\) with \(\langle I\cap R^{\bullet}\rangle\); equality is the local test. If the goal is to classify the Ring, explain why equality holds for every regular ideal, or exhibit one regular ideal where it fails. A single successful local calculation cannot discharge the universal claim.

The positive examples show two proof routes. For \(\mathbb Z\times\mathbb Z\), use the published finite-product result to classify the ring, then calculate a representative ideal to see how coordinatewise regularity works.^[1] For \(k[x,y]/(xy)\), establish Noetherianity using the polynomial ring's Hilbert basis theorem and passage to quotients, then apply the cited Noetherian-implies-Marot proposition.^[1] The elementary quotient calculation below illustrates a regular element but does not replace that universal theorem.

Knowledge Transfer¶

The test transfers literally among commutative unital rings because the roles keep the same meanings: regular element, regular ideal, and ideal generation. The two examples vary the construction of the carrier while preserving that exact test. One may use a class theorem to transport the conclusion to another qualifying ring, but only after proving the theorem's hypotheses for that ring.

Outside ring theory, phrases such as “generate from admissible elements” can be useful analogies, but they do not retain the algebraic meaning of non-zero-divisor or ring ideal. The broader catalog Prime Classification describes assigning objects to a class by rules; it helps explain the act of recognizing a Marot ring, but it is not the Marot condition itself. The named identity remains domain-specific.

Examples¶

Canonical: the product ring \(\mathbb Z\times\mathbb Z\)¶

The cited paper identifies finite products of domains as Marot rings and proves a product criterion.^[1] The ring \(\mathbb Z\times\mathbb Z\) is our concrete instance: multiplication is coordinatewise, so a pair is regular exactly when both coordinates are nonzero.^[1] The ideal \(2\mathbb Z\times3\mathbb Z\) is generated by the regular pair \((2,3)\), since multiplying it by arbitrary \((m,n)\) gives \((2m,3n)\).^[1] These calculations are deductions from the definition, not examples worked by the paper. They illustrate one regular ideal; the published class result establishes the ring-wide conclusion.

Mapped back: commutative unital ring → the coordinatewise product; regular-element filter → pairs with two nonzero coordinates; qualifying ideal → \(2\mathbb Z\times3\mathbb Z\), which contains \((2,3)\); universal generation equality → the displayed ideal works directly, and the finite-product result covers every regular ideal.

Applied: the quotient \(k[x,y]/(xy)\)¶

Let \(k\) be a field and \(R=k[x,y]/(xy)\). This quotient has nonzero zero divisors \(x\) and \(y\), whose product is zero.^[1] It is Noetherian as a quotient of the Noetherian polynomial ring \(k[x,y]\); hence the cited Proposition 3.3(1) makes it Marot.^[1] To see a regular element, map a polynomial class to its restrictions \((f(x,0),f(0,y))\) in \(k[x]\times k[y]\). The kernel is \((x)\cap(y)=(xy)\), so this map is injective. The class of \(x+y\) maps to \((x,y)\), whose two coordinates are nonzero in domains, and is therefore a non-zero-divisor.^[1] Its principal ideal is one qualifying ideal generated by its own regular member. These construction and ideal calculations are explicit deductions, not worked examples in the cited paper; the Noetherian class result is the ring-wide warrant.

Mapped back: commutative unital ring → \(k[x,y]/(xy)\); regular-element filter → non-zero-divisors including \(x+y\); qualifying ideal → \((x+y)\); universal generation equality → this principal ideal works directly, while Noetherianity plus Proposition 3.3(1) covers all qualifying ideals.

Structural Tensions¶

The defining property is a formal yes-or-no condition, rather than a balance between two opposed goals. Using arbitrary generators versus restricting to regular generators is the definition's discriminating test, not a further structural tension. An application may introduce its own competing objectives, but none belongs to every Marot ring as such.

Structural–Framed Character¶

Spectrum placement: strongly structural within commutative algebra, but domain-bound on the cross-domain spectrum. Evaluative weight: low; the condition classifies a ring by proof, without praising one algebraic structure over another. Human-practice dependence: low for the truth of the property, though the choice to study regular ideals is a mathematical research practice. Institutional origin: the name and published theorems belong to algebraic scholarship, while the defining equality can be tested independently of an institution or author. Vocabulary travel: “regular,” “ideal,” and “generation” have other meanings elsewhere; the complete test travels only where ring multiplication, zero divisors, and ideal generation remain literal. Import versus recognition: one recognizes the class by checking its defining equality or a covering theorem, rather than importing its name because another system seems to use “regular” parts. The portable skeleton is only a broad rule-based classification question already represented by the Prime Classification; the Marot property is the algebraic differentia. Its character: an exact, reusable class of commutative rings, framed by algebraic primitives rather than a cross-domain Prime.

Structural Core vs. Domain Accent¶

Skeletal relation: a class can be defined by a universal condition over a selected family of subobjects, using an allowed subset to generate each subobject. At that level the discussion resembles rule-based Classification, but the resemblance does not prove a typed Prime parent for this ring identity. Domain-bound mechanism: “subobject” means a ring ideal, “allowed” means non-zero-divisor, and “generate” means ideal generation by ring-linear combinations. Those operations are indispensable to the Marot test. Why domain-specific: removing them leaves a general classification schema but no criterion for being a Marot ring. The named class therefore retains literal reach only in commutative algebra; cross-domain versions would be analogies or different abstractions.

This entry is a kind of Commutative ring.

Classification is a real catalog Prime for assigning entities to categories by explicit rules. Calling a ring Marot is an instance of doing classification, but classification does not specify which ring ideals qualify or how their regular elements generate them. Constraint is another real Prime for a condition restricting admissible cases, yet every defined mathematical class can be described that way. The additional algebraic equality, not the generic fact of a constraint, distinguishes Marot rings. Closure concerns outputs of operations staying within a set; ideal absorption has a closure aspect, but the Marot differentia is generation by regular elements, not closure of an ideal under multiplication. These Primes are explanatory neighbors, not claimed canonical parents.

Relationships to Other Abstractions¶

Current abstraction Marot ring Domain-specific

Parents (1) — outgoing typed relations

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

Show alternative path (1)

Neighborhood in Abstraction Space¶

Marot ring sits in a moderately populated region (45^th 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 universal generation condition on its regular ideals. A Marot ring is a commutative ring with that further property.
Ring ideal: is one subobject of a ring. A Marot ring is a class of whole rings selected by a condition on all qualifying ideals.
Regular ideal (modular sense): the live catalog entry centers a right ideal with a modularity witness, including nonunital settings. The “regular ideal” in the Marot definition instead contains a non-zero-divisor.
v-Marot ring: uses a related divisorial-ideal condition. Ordinary Marot implies v-Marot in the cited paper; do not swap the two recognition rules.^[1]
Noetherian ring: satisfies an ascending-chain or finite-generation condition and is Marot by the cited proposition, but that sufficient condition is not the definition of Marot.^[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 (ring, regularity, Marot definition and product class), p. 7 Proposition 3.3(1) (Noetherian and v-Marot implications), and p. 16 Proposition 4.7(1) (product criterion). The two concrete ring-and-ideal calculations above are deductions using these statements, not examples printed in the paper. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27