Unique factorisation rings

Chatters, A. W.; Gilchrist, Martin; Wilson, David C.

doi:10.1017/s0013091500005526

Cited by 17 publications

(55 citation statements)

References 14 publications

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“…Chatters and Jordan studied in [9] a class of Noetherian (= left and right Noetherian) rings which in the commutative case consists of the Noetherian unique factorisation domains:…”

Section: Unique Factorisation Ringsmentioning

confidence: 99%

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A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

Venjakob¹,

Vogel²

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. In this paper and a forthcoming joint one with Y. Hachimori [15] we study Iwasawa modules over an infinite Galois extension k ∞ of a number field k whose Galois group G = G(k ∞ /k) is isomorphic to the semidirect product of two copies of the p-adic integers Z p . After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group X nr of the p-Hilbert class field over k ∞ . The main tool we use is a version of the Weierstrass preparation theorem, which we prove for certain skew power series with coefficients in a not necessarily commutative local ring. One striking result in our work is the discovery of the abundance of faithful torsion Λ(G)-modules, i.e. non-trivial torsion modules whose global annihilator ideal is zero. Finally we show that the completed group algebra F p [[G]] with coefficients in the finite field F p is a unique factorization domain in the sense of Chatters [8].

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“…Chatters and Jordan studied in [9] a class of Noetherian (= left and right Noetherian) rings which in the commutative case consists of the Noetherian unique factorisation domains:…”

Section: Unique Factorisation Ringsmentioning

confidence: 99%

“…Such elements are called normal and if I is a prime ideal in addition than a is called prime element. The class of UFRs contains the following much more restricted one (Remark (5) in [9]):…”

Section: Every Height-1 Prime Ideal Of a Is Principalmentioning

confidence: 99%

A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

Venjakob¹,

Vogel²

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…It was shown in the proof of Theorem 4.19 of [5] that K, = R. We have byK c X, so that yK c H and K c y~xH. [4] Unique factorisation in P. I. group-rings 235 PROOF.…”

Section: Preliminariesmentioning

confidence: 92%

“…Returning to the proof of Step (c), we are supposing that M n {FG) is a U. F. R. Therefore M n (FG) is an M-ring, by [5,Corollary 4.8]. It follows by routine arguments that FG is also an M-ring.…”

Section: 9]) It Follows Easily From This Thatmentioning

confidence: 98%

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Unique factorisation in P.I. group-rings

Chatters¹

1995

J Aust Math Soc A

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Height-One Prime Ideals in Semigroup Algebras Satisfying a Polynomial Identity

Jespers

Wang

2002

Journal of Algebra

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For a submonoid S of a torsion-free abelian-by-finite group, we describe the height-one prime ideals of the semigroup algebra K S . As an application we investigate when such algebras are unique factorization rings.  2002 Elsevier Science (USA) Let S be a submonoid of a group. In this paper we investigate when prime semigroup algebras K S are unique factorization rings that satisfy a polynomial identity. In the case S is a group, Chatters [4] showed that this is the case precisely when S is a dihedral free group which satisfies the ascending chain condition on cyclic subgroups. Chatters' result is fundamentally based on an earlier result of Brown [2], who characterized group algebras of polycyclic-by-finite groups that are unique factorization rings. The mentioned results can be extended to arbitrary unique factorization coefficient rings (see [1]). In the case S is a commutative monoid, Gilmer in [11] described when K S is a unique factorization domain. This paper

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Unique factorisation rings

Cited by 17 publications

References 14 publications

A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory

Unique factorisation in P.I. group-rings

Height-One Prime Ideals in Semigroup Algebras Satisfying a Polynomial Identity

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