关于E_ω-投射模和E_ω-内射模(英文)

On E_ω-projective Modules and E_ω-injective Modules

左R-模M称为Eω-内射模,如果对环R中任意的ω阶Euclid理想I来说,任何R-模同态能够拓展为R-模同态。左R-模M称为Eω-投射模,若对环R中任意的ω阶Euclid理想I和任何R-模同态f∈HomR(M,R/I),存在R-模同态g∈HomR(M,R)使得f=πg,其中π是自然同态。本文证明P和Q均是Eω-投射模当且仅当PQ是Eω-投射模。进而,又证明了每一个左R-模是Eω-投射的当且仅当每一个左R-模是Eω-内射。

A left R-module M is called Eω-injective if for any ω-stage Euclidean ideal I of R,any R-module homomorphism may be extended to an R-module homomorphism.A left R-module M is called Eω-projective if for any ω-stage Euclidean ideal I of R and any R-module homomorphism f∈HomR（M,R/I）,there exists an R-module homomorphism g∈HomR（M,R）such that f=πg where π is the canonical epimorphism.We prove,in the article,that both P and Q are Eω-projective if and only if so is PQ.Further,we show that every left R-module is Eω-projective if and only if every left R-module is Eω-injective.