Hereditary Noetherian Prime Rings and Idealizers

Levy, Lawrence S.; Robson, J. C.

doi:10.1090/surv/174

Cited by 12 publications

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“…By a monoid we mean a cancellative semigroup with identity. As a general reference for noncommutative Noetherian rings we use [ MR01 MR01], for HNP rings [ LR11 LR11].…”

Section: Background and Notationmentioning

confidence: 99%

“…By an (R-)module, without further qualification, we mean a right R-module. We will make use of the theory of finitely generated projective modules over an HNP ring, as given by Levy and Robson in [ LR11 LR11]. Recall that two projective modules P and Q are stably isomorphic if there exists an n ∈ N 0 such that P ⊕ R n ∼ = Q ⊕ R n .…”

Section: Background and Notationmentioning

confidence: 99%

“…If R is commutative, it is easily seen that C(R) is isomorphic to the ideal class group as it is traditionally defined. The main result in this section is Theorem 3.15 3.15, which shows that C(R) is isomorphic to the ideal class group G(R) as defined in [LR11 LR11] as a direct summand of K 0 (R). Moreover, we show that the distinguished subset G max (R) ⊂ G(R), which appears in the construction of a transfer homomorphism, is preserved under Morita equivalence and passage to a Dedekind right closure.…”

Section: Introductionmentioning

confidence: 97%

“…For the more general class of HNP rings, the multiplicative ideal theory of two-sided ideals has been investigated. We refer the reader to [ LR11 LR11,§22], and also mention [Rum01 Rum01, AM16 AM16, RY16 RY16] for samples of recent progress in noncommutative multiplicative ideal theory. However, a one-sided ideal theory seems not to have been developed.…”

Section: Introductionmentioning

confidence: 99%

“…We make extensive use of the structure theory of finitely generated projective modules over HNP rings, which can be viewed as a far-reaching generalization of Steinitz's theorem. This theory was developed over the last decades, chiefly by Eisenbud, Levy, and Robson, and is presented in the monograph [ LR11 LR11].…”

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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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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“…By a monoid we mean a cancellative semigroup with identity. As a general reference for noncommutative Noetherian rings we use [ MR01 MR01], for HNP rings [ LR11 LR11].…”

Section: Background and Notationmentioning

confidence: 99%

Section: Background and Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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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Factorizations of Elements in Noncommutative Rings: A Survey

Smertnig

2016

Springer Proceedings in Mathematics &Amp; Statistics

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We survey results on factorizations of non-zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.

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Characterizing Rings in Terms of the Extent of the Injectivity of Their Simple Modules

Saraç

Aydoğdu

2023

Mediterr. J. Math.

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Hereditary Noetherian Prime Rings and Idealizers

Cited by 12 publications

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Factorizations in Bounded Hereditary Noetherian Prime Rings

Factorizations in Bounded Hereditary Noetherian Prime Rings

Factorizations of Elements in Noncommutative Rings: A Survey

Characterizing Rings in Terms of the Extent of the Injectivity of Their Simple Modules

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