Multiplicative Ideal Theory in Noncommutative Rings

Akalan, Evrim; Marubayashi, Hidetoshi

doi:10.1007/978-3-319-38855-7_1

Cited by 8 publications

(10 citation statements)

References 59 publications

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“…(2) Recall that P/Q has finite length if and only if udim P = udim Q by [26,Corollary 12.17]. Then the claim follows from (1) and A cyclic module R/I with I a right ideal of R has finite length if and only if I is a right R-ideal. In particular, a cyclically presented module, that is, a module of the form R/aR with a ∈ R, has finite length if and only if a ∈ R • .…”

Section: Proposition 32mentioning

confidence: 98%

“…For the more general class of HNP rings, the multiplicative ideal theory of two-sided ideals has been investigated. We refer the reader to [26, § 22], and also mention [1,31,32] for samples of recent progress in non-commutative multiplicative ideal theory. However, no one-sided ideal theory seems to have been developed.…”

Section: Introductionmentioning

confidence: 99%

“…C(R) = { R/I | I a right R-ideal in the principal genus }.Proof (1). We proceed by induction on n. If n = 0, let J = I.…”

mentioning

confidence: 99%

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Factorizations in Bounded Hereditary Noetherian Prime Rings

Smertnig¹

2018

Proceedings of the Edinburgh Mathematical Society

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If H is a monoid and a = u 1 · · · u k ∈ H with atoms (irreducible elements) u 1 , . . . , u k , then k is a length of a, the set of lengths of a is denoted by L(a), and L(H) = { L(a) | a ∈ H } is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R • can be written as a product of atoms. We show that, if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R • to the monoid B of zero-sum sequences over a subset G max (R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R • and B coincide. It is well-known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.The principal aim of factorization theory is to describe the various phenomena of non-uniqueness of factorizations by suitable arithmetical invariants (such as sets of lengths), and to study the interaction of these arithmetical invariants with classical algebraic invariants of the underlying ring (such as class groups). Factorization theory has its origins in algebraic number theory and in commutative algebra. The focus has been on commutative Noetherian domains, commutative Krull domains and monoids, their monoids of ideals, and their monoids of modules. We refer to [And97 And97, Nar04 Nar04, Cha05 Cha05, GHK06 GHK06, Ger09 Ger09, BW13 BW13, Fac12 Fac12, LW12 LW12, BW13 BW13, FHL13 FHL13, CFGO16 CFGO16] for recent monographs, conference proceedings, and surveys on the topic.A key strategy, lying at the very heart of factorization theory, is the construction of a transfer homomorphism from a ring or monoid of interest to a simpler one. This allows one to study the arithmetic of the simpler object and then pull back the information to the original object of interest. An exposition of transfer principles in the commutative setting can be found in [GHK06 GHK06, Section 3.2]. The best investigated class of simpler objects occurring in this context is that of monoids of zero-sum sequences over subsets of abelian groups. These monoids are commutative Krull monoids having a combinatorial flavor; questions about factorizations are reduced to questions about zero-sum sequences, which are studied with methods from additive combinatorics. We refer the reader to the survey [Ger09 Ger09] for the interplay between additive combinatorics and the arithmetic of monoids.Until relatively recently, the study of factorizations of elem...

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Section: Proposition 32mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Factorizations in Bounded Hereditary Noetherian Prime Rings

Smertnig¹

2018

Proceedings of the Edinburgh Mathematical Society

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“…Thus, every commutative Krull domain is a normalizing Krull ring. For examples and background on non-commutative (normalizing) Krull rings we refer to [53,10,45,1], and for background on factorizations in the non-commutative setting to [4,52]. In particular, normalizing Krull monoids are transfer Krull.…”

Section: Thenmentioning

confidence: 99%

A characterization of length-factorial Krull monoids

Geroldinger,

Zhong

2021

Preprint

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“…This then yields a class of non-Noetherian algebras that have a nice arithmetical structure. For more information on Krull orders and Krull monoids, we refer the reader to the monographs [5,8] and to some older and recent papers [1,2,4,6,9].…”

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confidence: 99%

Krull orders in nilpotent groups: corrigendum and addendum

Jespers

Okniński

2016

Arch. Math.

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Noncommutative Krull domains that are determined by submonoids of torsion-free nilpotent groups are investigated. A complete description is given in case the group G is nilpotent of class two and its abelianisation is torsion-free and satisfies the ascending chain condition on cyclic subgroups. The result corrects and extends an earlier result by the authors to the case that G is not necessarily finitely generated and yields a class of non-Noetherian algebras that have a nice arithmetical structure.In [7] the authors investigate the arithmetical properties, such as being a Krull order, of a class of algebras that are non-Noetherian, which is motivated by deformations of some quadratic algebras showing up in the classification of four dimensional noncommutative projective surfaces, see [13,14]. The algebras under consideration turn out to be semigroup algebras K[S] over a field K, where S is a submonoid of a torsion-free nilpotent group of class 2.For definitions on maximal orders and Krull orders, we refer to [7,11]. So, a prime (left and right) Goldie ring R with classical ring of quotients Q is said to be a Krull order in Q if it is a maximal order that satisfies the ascending chain condition on integral divisorial ideals.The main result in [7], namely Theorem 3.4, is a characterization of when such an algebra K[S] is a Krull order (note that it is a domain as S is a submonoid of a torsion-free nilpotent group). It is shown that if

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Multiplicative Ideal Theory in Noncommutative Rings

Cited by 8 publications

References 59 publications

Factorizations in Bounded Hereditary Noetherian Prime Rings

Factorizations in Bounded Hereditary Noetherian Prime Rings

A characterization of length-factorial Krull monoids

Krull orders in nilpotent groups: corrigendum and addendum

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