Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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w-MATLIS COTORSION MODULES AND w-MATLIS DOMAINS

  • Pu, Yongyan (College of Mathematics and Software Science Sichuan Normal University) ;
  • Tang, Gaohua (College of Sciences Beibu Gulf University) ;
  • Wang, Fanggui (College of Mathematics and Software Science Sichuan Normal University)
  • Received : 2018.10.01
  • Accepted : 2019.02.08
  • Published : 2019.09.30
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Abstract

Let R be a domain with its field Q of quotients. An R-module M is said to be weak w-projective if $Ext^1_R(M,N)=0$ for all $N{\in}{\mathcal{P}}^{\dagger}_w$, where ${\mathcal{P}}^{\dagger}_w$ denotes the class of GV-torsionfree R-modules N with the property that $Ext^k_R(M,N)=0$ for all w-projective R-modules M and for all integers $k{\geq}1$. In this paper, we define a domain R to be w-Matlis if the weak w-projective dimension of the R-module Q is ${\leq}1$. To characterize w-Matlis domains, we introduce the concept of w-Matlis cotorsion modules and study some basic properties of w-Matlis modules. Using these concepts, we show that R is a w-Matlis domain if and only if $Ext^k_R(Q,D)=0$ for any ${\mathcal{P}}^{\dagger}_w$-divisible R-module D and any integer $k{\geq}1$, if and only if every ${\mathcal{P}}^{\dagger}_w$-divisible module is w-Matlis cotorsion, if and only if w.w-pdRQ/$R{\leq}1$.

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Acknowledgement

Supported by : National Natural Science Foundation of China, Guangxi Natural Science Foundation

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