分次Matlis余挠模与分次Matlis整环

Graded Matlis cotorsion modules and graded Matlis domains

设R是G-分次整环。本文引入了分次h-可除R-模,分次Matlis余挠R-模与分次Matlis整环的概念。证明了:(1)设M是分次模,则gr-pdR(M)≤1当且仅当对任何分次h-可除模D,有EXTR^1(M,D)=0;(2)M是分次Matlis余挠模当且仅当对任何σ∈G,M(σ)是分次Matlis余挠模;(3)R是分次Matlis整环当且仅当分次投射维数不超过1的分次模类与分次h-可除模类构成一个分次余挠理论。

Let R be a G-graded integral domain. The notions of graded h -divisible R -module, graded Matlis cotorsion R -module and graded Matlis domain are introduced. It is shown in this paper that:(1)If M is a graded module, then gr-pd R(M)≤1 if and only if EXT^1R(M,D)=0 for each graded h -divisible module D;(2)M is a graded Matlis cotorsion module if and only if M(σ) is a graded Matlis cotorsion module for each σ∈G;(3)R is a graded Matlis domain if and only if the pair ( gr-P1, gr-LC) forms a graded cotorsion theory, where gr-P1 is the class of graded modules of graded projective dimension at most one and gr-LC is the class of graded h -divisible modules.