Journal of the
Korean Mathematical Society JKMS

Let $R$ be a ring in which $Nil(R)$ is a divided prime ideal of $R$. Then, for a suitable property $X$ of integral domains, we can define a $\phi$-$X$-ring if $R/Nil(R)$ is an $X$-domain. This device was introduced by Badawi \cite{Ayman1} 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 $\phi$-Schreier rings, $\phi$-quasi-Schreier rings, $\phi$-almost-rings, $\phi$-almost-quasi-Schreier rings, $\phi$-$GCD$ rings, $\phi$-generalized $GCD$ rings and $\phi$-almost $GCD$ rings as rings $R$ with $Nil(R)$ a divided prime ideal of $R$ such that $R/Nil(R)$ 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 $x, y, z\in D\backslash 0,$ $x\mid yz$ in $D$ implies that $x=rs$ where $r\mid y$ and $s\mid z$. An integrally closed pre-Schreier domain was initially called a Schreier domain by Cohn in \cite{Cohn} where it was shown that a GCD domain is a Schreier domain.

Keywords: $\phi$-primal, $\phi$-Schreier ring, $\phi$-quasi-Schreier ring, $\phi$-$GCD$ ring

MSC numbers: Primary 16N99, 16S99; Secondary 06C05, 16N20