Rings with a theory of greatest common divisors
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''.
scientific article; zbMATH DE number 1135229