扩环上的投射模

Projective Modules over Extension Rings

研究了环的投射模在扩环上的遗传特征，得到了：设ＲＳ，Ｃ（Ｒ，Ｓ）为Ｓ的极大理想，则有：（１）若Ｒ∈ＰＦ，则Ｓ∈ＰＦ；（２）若Ｒ［ｘ１，…，ｘｎ］∈ＰＦ，则Ｓ［ｘ１，…，ｘｎ］∈ＰＦ，（３）若Ｒ［［ｘ１，…，ｘｎ］］∈ＰＦ，则Ｓ［［ｘ１，…，ｘｎ］］∈ＰＦ．

in this paper,by virtue of Milnor sequare and K0 group,we obtain the following results: Let R S,the conductor C(R,S) be a maximum ideal of S. (1)If every finitely generated projective R-module is free, then so is eVery finitely generated projective module over S. (2)If every finitely generated projective R[x1,...,xn]-module is free,then so is every finitely generated projective module over S[x1...,xn]. (3)If every finitely generated projective R[[x1,...,xn]]-module is free,then so is every finitely generated projective module over S[[x1, ...,xn]].