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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  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 anddoi:10.4134/jkms.j150380 fatcat:a6a6a5ejp5fsvjvurtp2agwlv4