Ahmad Yousefian Darani, Mahdi Rahmatinia
2016 Journal of the Korean Mathematical Society  
Let R be a ring in which N il(R) is a divided prime ideal of R. Then, for a suitable property X of integral domains, we can define a φ-X-ring if R/N il(R) is an X-domain. This device was introduced by Badawi [8] to study rings with zero divisors with a homomorphic image a particular type of domain. We use it to introduce and study a number of concepts such as φ-Schreier rings, φ-quasi-Schreier rings, φ-almost-rings, φ-almost-quasi-Schreier rings, φ-GCD rings, φ-generalized GCD rings and
more » ... GCD rings as rings R with N il(R) a divided prime ideal of R such that R/N il(R) is a Schreier domain, quasi-Schreier domain, almost domain, almost-quasi-Schreier domain, GCD domain, generalized GCD domain and almost GCD domain, respectively. We study some generalizations of these concepts, in light of generalizations of these concepts in the domain case, as well. Here a domain D is pre-Schreier if for all x, y, z ∈ D\0, x | yz in D implies that x = rs where r | y and s | z. An integrally closed pre-Schreier domain was initially called a Schreier domain by Cohn in [15] where it was shown that a GCD domain is a Schreier domain.
doi:10.4134/jkms.j150380 fatcat:a6a6a5ejp5fsvjvurtp2agwlv4