环的有限正规扩张和模

Finite Normal Extensions and Modules of Rings

本文证明了:若S是R的一个有限正规扩张,则(1)_RF是平坦的,当且仅当S(?)_RF是一个平坦的左S-模;(2)有限生成模P_R是投射的,当且仅当P(?)_RS是一个投射的右S-模。 若S是R的一个右自由有限正规扩张,则P_R是投射的,当且仅当P(?)_RS是一个投射的右S-模。 并应用这些结果于“从R的一个有限正规扩张S具有某种性质去断定R也具有该种性质”。得到了一些新的结果。

The present paper reports that if S is a finite normal extension of R, then (1) _RF is flat iff as a left S-module is flat, (2) Finitely generated module p_R is projective iff as a right Smodule is projective; if S is a right free finite normal extension of R, then p_R is projective iff as a right S-module is projective.Using the same method as Laila Soueif(cf. [2]1.3), the above results can be applied to judze whether R has this property. We have judged whether R has such properties on the basis of properties S, a finite normal extension of R has. We concluded that if S, a finite normalizing extension of R is a Von Neumann regular ring, so is R, then. This result not only improves [4] 3.3, but also deduces [4] 3.5 directly witlout use of Parmenter and Stewart's proof.