Converse Prime Element Theorems for Arithmetical Semigroups

We define the cover of an RG-module V to consist of an RG lattice Ṽ and a homomorphism π : Ṽ→ V such that π induces an isomorphism on Ext*RG(M, —) for any RG-lattice M. Here G is a finite group and, for simplicity in this introduction, R is a complete discrete valuation ring of characteristic zero with prime element p and perfect valuation class field. Let pn(G) be the highest power of p that divides |G| and, given an RG-lattice M, let pn(M) be the smallest power of p such that pn(M) idM : M→M factors through a projective lattice: n(M)≦n(G).

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Recent results on weakly factorial domains

ITM Web of Conferences ◽

10.1051/itmconf/20182001001 ◽

2018 ◽

Vol 20 ◽

pp. 01001

Author(s):

Chang Gyu Whan

Keyword(s):

Polynomial Ring ◽

Quotient Group ◽

Integral Domain ◽

Laurent Polynomial ◽

Semigroup Ring ◽

Prime Element ◽

Cancellative Monoid ◽

Factorial Domain ◽

Laurent Polynomial Ring

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

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Unique factorisation rings

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500005526 ◽

1992 ◽

Vol 35 (2) ◽

pp. 255-269 ◽

Cited By ~ 5

Author(s):

A. W. Chatters ◽

M. P. Gilchrist ◽

D. Wilson

Keyword(s):

Prime Ideal ◽

Prime Ring ◽

Polynomial Identity ◽

Noetherian Ring ◽

Basic Theory ◽

Maximal Order ◽

Prime Element ◽

Polynomial Extension ◽

Simple Ring ◽

Left And Right

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

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