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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.$w$-$\pd_RQ/R\leq1$.
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참고문헌 (24)
참고문헌 신청
F. A. A. Almahdi / 2018 / On the right orthogonal complement of the class of w-at modules / J. Ramanujan Math. Soc 33(2):159~175
S. Bazzoni / 2018 / S-almost perfect commutative rings / Preprint arXiv: 1801. 04820
S. Bazzoni / 2004 / On strongly at modules over integral domains / Rocky Moun-tain J. Math 34(2):417~439
L. Fuchs / 2001 / Modules over Non-Noetherian Domains / American Mathematical Society
I. Kaplansky / 1966 / The homological dimension of a quotient eld / Nagoya Math. J 27:139~142
S. Bazzoni / 2018 / S-almost perfect commutative rings / Preprint arXiv: 1801. 04820
S. Bazzoni / 2004 / On strongly at modules over integral domains / Rocky Moun-tain J. Math 34(2):417~439
L. Fuchs / 2001 / Modules over Non-Noetherian Domains / American Mathematical Society
I. Kaplansky / 1966 / The homological dimension of a quotient eld / Nagoya Math. J 27:139~142
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