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Cohen-Kaplansky domains: Integral domains with a finite number of irreducible elements. (English) Zbl 0773.13005

From the text: “We define an integral domain \(R\) to be a Cohen-Kaplansky domain (CK domain) if every element of \(R\) is a finite product of irreducible elements and \(R\) has only finitely many nonassociate irreducible elements. The purpose of this paper is to investigate CK domains. Many conditions equivalent to \(R\) being a CK domain are given, for example, \(R\) is a CK domain if and only if \(R\) is a one-dimensional semilocal domain and for each nonprincipal maximal ideal \(M\) of \(R\), \(R/M\) is finite and \(R_ M\) is analytically irreducible, or, if and only if \(G(R)\), the group of divisibility of \(R\), is finitely generated and rank \(G(R)=|\text{Max}(R)|\). We show that a CK domain is a certain special type of composite or pullback of a subring of a finite homomorphic image of a semilocal PID. Noetherian domain with \(G(R)\) finitely generated are also investigated”.

MSC:

13G05 Integral domains
13A05 Divisibility and factorizations in commutative rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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References:

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