The class semigroup of orders in number fields

Zanardo, Paolo; Zannier, Umberto

doi:10.1017/s0305004100072170

Cited by 36 publications

(32 citation statements)

References 2 publications

(2 reference statements)

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“…Bazzoni and Salce in [1] study ideal class semigroups of valuation domains and later Bazzoni in [2] studies the structure of ideal class semigroups of Prüfer domains. Ideal class semigroups of orders in number fields are investigated by Zanardo and Zannier in [21].…”

Section: Note That S(r) Is a Commutative Semigroup With Identity [R]mentioning

confidence: 99%

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Ideal class semigroups of overrings

Sega

2007

Journal of Algebra

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Let R be an integral domain an let T be an overring of R. There is a canonical semigroup homomorphism between the ideal class semigroup of R and the ideal class semigroup of T . We investigate conditions under which this semigroup homomorphism is surjective and we apply the results we obtain to the study of overrings of Clifford regular domains. We recover some known results of Bazzoni and we prove in certain more general situations that the Clifford regular property is inherited by an overring. In particular, we prove that if R is a Clifford regular domain such that the integral closure of R is a fractional overring, then every overring of R is Clifford regular. We also characterize among Clifford regular domains the ones that are stable.

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Section: Note That S(r) Is a Commutative Semigroup With Identity [R]mentioning

confidence: 99%

“…G [L] Clifford regular domains are introduced by Zanardo and Zannier in [21] and Bazzoni and Salce in [1]. Bazzoni in [3] is the first to write down the definition of Clifford regular domains and study them in greater detail.…”

Section: Note That S(r) Is a Commutative Semigroup With Identity [R]mentioning

confidence: 99%

Ideal class semigroups of overrings

Sega

2007

Journal of Algebra

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“…There is only one maximal ideal lying over P in R if p is ramified or inert. By [12,Proposition 12], we have P = pZ + nωZ when p|n.…”

Section: The Quadratic Casementioning

confidence: 99%

Cale Bases in Algebraic Orders

Picavet-L’Hermitte¹

2003

Ann. math. Blaise Pascal

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Cale Bases in Algebraic OrdersMartine Picavet-L'Hermitte AbstractLet R be a non-maximal order in a finite algebraic number field with integral closure R. Although R is not a unique factorization domain, we obtain a positive integer N and a family Q (called a Cale basis) of primary irreducible elements of R such that x N has a unique factorization into elements of Q for each x ∈ R coprime with the conductor of R. Moreover, this property holds for each nonzero x ∈ R when the natural map Spec(R) → Spec(R) is bĳective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.

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“…Zanardo and Zannier proved that all orders in quadratic fields are Clifford regular domains [20] while Bazzoni and Salce showed that all valuation domains are Clifford regular [5]. The study of Clifford regular domains was then carried on by S. Bazzoni [1,2,3,4].…”

Section: Introductionmentioning

confidence: 99%