C-投射(内射,平坦)模与优越扩张

Homological dimensions with respect to semidualizing modules and excellent extensions

令C作为R-模为半对偶模,其中R为交换环。在(几乎)优越扩张的条件下研究了与半对偶模C相关模类的传递性,讨论了C-投射,内射及平坦预盖及预包的相关性质。作为应用,证明了当环扩张S≥R为优越扩张时,R为诺特环当且仅当S为诺特环;R为凝聚环当且仅当S为凝聚环。

Let C be a semidualizing R-module with R being a commutative ring. It is irivestigated the transfer properties of C-homological dimensions under （almost） excellent extensions, and it is discussed that the precovering and preenvel- oping properties of the C-projectives, C-injectives, and C-flats. As applications, it is proved that if S≥R is an excellent extension, then R is Noetherian if and only if S is Noetherian, and R is coherent if and only if S is coherent.