• 충청수학회지
• /
• 제25권4호
• /
• pp.711-715
• /
• 2012

In terms of divisible-like modules, some equivalent conditions for an integral domain R to be a generalized GCD domain are given.

• 충청수학회지
• /
• 제24권1호
• /
• pp.131-135
• /
• 2011

It is shown that generalized GCD domains satisfy a certain property of injectivity and conversely this property characterizes generalized GCD domains.

• 대한수학회지
• /
• 제53권5호
• /
• pp.1057-1075
• /
• 2016

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 [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 ${\phi}$-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 | 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.

• 대한수학회보
• /
• 제41권3호
• /
• pp.473-482
• /
• 2004

Let D be an integral domain, I a proper ideal of D, and R =D[It, $t^{-1}$] a generalized Rees ring, where t is an indeterminate. For suitable conditions, we show that R satisfies the ACCP (resp., is a BFD, an FFD, a (pre-) Schreier domain, a G-GCD domain, a PVMD, a v-domain) if and only if D satisfies the ACCP (resp., is a BFD, an FFD, a (pre-) Schreier domain, a G-GCD domain, a PVMD, a v-domain).