• Title/Summary/Keyword: GCD-domain

Search Result 12, Processing Time 0.017 seconds

ON 𝜙-SCHREIER RINGS

  • Darani, Ahmad Yousefian;Rahmatinia, Mahdi
    • Journal of the Korean Mathematical Society
    • /
    • v.53 no.5
    • /
    • 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.

KAPLANSKY-TYPE THEOREMS IN GRADED INTEGRAL DOMAINS

  • CHANG, GYU WHAN;KIM, HWANKOO;OH, DONG YEOL
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.4
    • /
    • pp.1253-1268
    • /
    • 2015
  • It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky's theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, $B{\acute{e}}zout$ domain, valuation domain, Krull domain, ${\pi}$-domain).

SEMISTAR G-GCD DOMAIN

  • Gmiza, Wafa;Hizem, Sana
    • Journal of the Korean Mathematical Society
    • /
    • v.56 no.6
    • /
    • pp.1689-1701
    • /
    • 2019
  • Let ${\star}$ be a semistar operation on the integral domain D. In this paper, we prove that D is a $G-{\tilde{\star}}-GCD$ domain if and only if D[X] is a $G-{\star}_1-GCD$ domain if and only if the Nagata ring of D with respect to the semistar operation ${\tilde{\star}}$, $Na(D,{\star}_f)$ is a G-GCD domain if and only if $Na(D,{\star}_f)$ is a GCD domain, where ${\star}_1$ is the semistar operation on D[X] introduced by G. Picozza [12].

SOME EXAMPLES OF ALMOST GCD-DOMAINS

  • Chang, Gyu Whan
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.24 no.3
    • /
    • pp.601-607
    • /
    • 2011
  • Let D be an integral domain, X be an indeterminate over D, and D[X] be the polynomial ring over D. We show that D is an almost weakly factorial PvMD if and only if D + XDS[X] is an integrally closed almost GCD-domain for each (saturated) multiplicative subset S of D, if and only if $D+XD_1[X]$ is an integrally closed almost GCD-domain for any t-linked overring $D_1$ of D, if and only if $D_1+XD_2[X]$ is an integrally closed almost GCD-domain for all t-linked overrings $D_1{\subseteq}D_2$ of D.

THE CLASS GROUP OF D*/U FOR D AN INTEGRAL DOMAIN AND U A GROUP OF UNITS OF D

  • Chang, Gyu Whan
    • Korean Journal of Mathematics
    • /
    • v.17 no.2
    • /
    • pp.189-196
    • /
    • 2009
  • Let D be an integral domain, and let U be a group of units of D. Let $D^*=D-\{0\}$ and ${\Gamma}=D^*/U$ be the commutative cancellative semigroup under aU+bU=abU. We prove that $Cl(D)=Cl({\Gamma})$ and that D is a PvMD (resp., GCD-domain, Mori domain, Krull domain, factorial domain) if and only if ${\Gamma}$ is a PvMS(resp., GCD-semigroup, Mori semigroup, Krull semigroup, factorial semigroup). Let U=U(D) be the group of units of D. We also show that if D is integrally closed, then $D[{\Gamma}]$, the semigroup ring of ${\Gamma}$ over D, is an integrally closed domain with $Cl(D[{\Gamma}])=Cl(D){\oplus}Cl(D)$; hence D is a PvMD (resp., GCD-domain, Krull domain, factorial domain) if and only if $D[{\Gamma}]$ is.

  • PDF

FACTORIZATION AND DIVISIBILITY IN GENERALIZED REES RINGS

  • Kim, Hwan-Koo;Kwon, Tae-In;Park, Young-Soo
    • Bulletin of the Korean Mathematical Society
    • /
    • v.41 no.3
    • /
    • 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).

t-SPLITTING SETS S OF AN INTEGRAL DOMAIN D SUCH THAT DS IS A FACTORIAL DOMAIN

  • Chang, Gyu Whan
    • Korean Journal of Mathematics
    • /
    • v.21 no.4
    • /
    • pp.455-462
    • /
    • 2013
  • Let D be an integral domain, S be a saturated multi-plicative subset of D such that $D_S$ is a factorial domain, $\{X_{\alpha}\}$ be a nonempty set of indeterminates, and $D[\{X_{\alpha}\}]$ be the polynomial ring over D. We show that S is a splitting (resp., almost splitting, t-splitting) set in D if and only if every nonzero prime t-ideal of D disjoint from S is principal (resp., contains a primary element, is t-invertible). We use this result to show that $D{\backslash}\{0\}$ is a splitting (resp., almost splitting, t-splitting) set in $D[\{X_{\alpha}\}]$ if and only if D is a GCD-domain (resp., UMT-domain with $Cl(D[\{X_{\alpha}\}]$ torsion UMT-domain).