Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXTENSIONS OF NAGATA'S THEOREM

  • Hamed, Ahmed (Department of Mathematics Faculty of Sciences)
  • Received : 2017.06.16
  • Accepted : 2017.10.25
  • Published : 2018.07.01
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Abstract

In [1], the authors generalize the concept of the class group of an integral domain $D(Cl_t(D))$ by introducing the notion of the S-class group of an integral domain where S is a multiplicative subset of D. The S-class group of D, $S-Cl_t(D)$, is the group of fractional t-invertible t-ideals of D under the t-multiplication modulo its subgroup of S-principal t-invertible t-ideals of D. In this paper we study when $S-Cl_t(D){\simeq}S-Cl_t(D_T)$, where T is a multiplicative subset generated by prime elements of D. We show that if D is a Mori domain, T a multiplicative subset generated by prime elements of D and S a multiplicative subset of D, then the natural homomorphism $S-Cl_t(D){\rightarrow}S-Cl_t(D_T)$ is an isomorphism. In particular, we give an S-version of Nagata's Theorem [13]: Let D be a Krull domain, T a multiplicative subset generated by prime elements of D and S another multiplicative subset of D. If $D_T$ is an S-factorial domain, then D is an S-factorial domain.

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References

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