内射强Precover

INJECTIVE STRONG PRECOVERS

1 引言设 R 是有单位元的结合环,我们约定:除了特别声明外,R-模均指右 R 模,Noethe-r 环指右 Noether 环,E(M)表示模 M 的内射包.设 M 是 R-模,E 是内射 R 模,根据 Enochs[1],E 以及 R-同态(?)∶E→M 叫 M的内射 Precover,如果对任意的内射模 E′及 R 同态(?)∶E′→M,都有 R-同态 f∶E′→E,使得(?)=(?)f.进一步称内射 Precover (?)∶E→M 为 M 的内射 Cover,如果使得(?)=(?)f 的同态 f∶E→E 只能是 E 的自同构.关于内射 Precover 和内射 Cover 的讨论,已有了大量的结果,如[1]、[4]、[5]等,在应用方面也出现了如[3]的结果.

An injective Precover(injective cover)(?):E→M of a module M is calledinjective strong precover(injective strong cover)of M if (?) is epimorphism,and an injective strong precovr (?):E→M of M is called K—indecomposableinjective strong precover of M if Ker (?) is indecomposable.In this paper,we have some work as following:(i) gave some necessary and sufficient conditions in which a R—moduleM have injective strong precover and injectve strong cover,or K—indecomp-osable injective strong precover.(iii) proved,in certain conditions,that there exist a long exact sequenceas following for a R—module M:…→SP_n(M)→…→SP_0(M)→M→0where SP_n(M)(n=0, 1, 2,…)are injective R—module determined uni(?)uelyby M.(iii)difined and studied C(M) and K(M) for a R—module M.