$w$-Matlis cotorsion modules and $w$-Matlis domains
Bull. Korean Math. Soc. 2019 Vol. 56, No. 5, 1187-1198
Published online September 30, 2019
Yongyan Pu, Gaohua Tang, Fanggui Wang
Panzhihua University; Beibu Gulf University; Sichuan Normal University
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 $N\in \mathcal{P}^{\dag}_w$, where $\mathcal{P}^{\dag}_w$ 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 $\leq1$. 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 $\mathcal{P}^{\dag}_w$-divisible $R$-module $D$ and any integer $k\geq1$, if and only if every $\mathcal{P}^{\dag}_w$-divisible module is $w$-Matlis cotorsion, if and only if w.$w$-$\pd_RQ/R\leq1$.
Keywords : $\mathcal{P}^{\dag}_w$-divisible modules, weak $w$-projective module, $w$-Matlis cotorsion module, $w$-strongly flat module, $w$-Matlis domain
MSC numbers : 13C11, 13C99, 13G05
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