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Rings with a theory of greatest common divisors. (English) Zbl 0901.13001

We quote the author’s words: “In this paper we introduce the concept of a ring with gcd-theory to generalize the notion of a gcd-domain. We call a domain \(D\) ring with gcd-theory if there is a homomorphism \((\cdot)\) from \(D^*\) into a gcd-monoid \({\mathcal G}^+\) such that the divisibility properties of \(D\) are faithfully transferred to \({\mathcal G}^+\) and two elements of \({\mathcal G}^+\) are equal if they divide the same elements \((a)\), \(a\in D^*\). By means of *-ideal theory we study the fundamental properties of rings with gcd-theory. We show that a ring \(D\) with gcd-theory is simply a \(\nu\)-domain. We examine the behaviour of rings with gcd-theory when passing on to the integral closure in algebraic field extensions. Finally, we take a closer look at special cases of gcd-theories, namely to rings with gcd-theory of finite type, i.e. rings with gcd-theory in which any element of \({\mathcal G}^+\) can be written as the greatest common divisor of a finite number of elements \((a)\), \(a\in D^*\), and to the well-known rings with divisor theory”.

MSC:

13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F07 Euclidean rings and generalizations
13A15 Ideals and multiplicative ideal theory in commutative rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
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