关于多项式环

On Polynomial Rings

证明了左Ｒ－模Ｍ是内射的当且仅当分次左Ｒ－模Ｍ是ｇｒ－内射的，当且仅当分次左Ｒ－模Ｍ是ｇｒ－内射的；左Ｒ－模Ｍ是Ｎｏｅｔｈｅｒ的当且仅当分次左Ｒ［ｘ］－模Ｍ［ｘ］是ｇｒ－Ｎｏｅｔｈｅｒ的，当且仅当分次左Ｒ［ｘ］－模Ｍ［ｘ］是Ｎｏｅｔｈｅｒ的；左Ｒ－模Ｍ是Ａｒｔｉｎ的当且仅当分次左Ｒ［ｘ］－模Ｍ［ｘ］是ｇｒ－Ａｒｔｉｎ的，当且仅当分次左Ｒ［ｘ］－模Ｍ［ｘ］是Ａｒｔｉｎ的；双模ＲＭＳ定义了一个拟对偶当且仅当分次双模Ｒ［ｘ］Ｍ［ｘ－１］Ｓ［ｘ］定义了一个ｇｒ－拟对偶．

Proves the following result:A left R module M is injective if and only if the graded left R module is gr injective if and only if graded left R module M is gr injective. A left R module M is Noetherian (Artinian) if and only if the graded left R module M is gr Noetherian (Artinian).An (R,S) bimodule RM S defines a quasi duality if and only if graded (R,S) bimodule R M[x -1 ] S defines a gr quasi duality.