广义τ-奇异模和τ-非奇异模

Generalizedτ-singular andτ-nonsingular modules

设τ是遗传挠理论,提出了广义τ-奇异和τ-非奇异的概念.称模M中的元素m是广义τ-奇异的,如果其右零化子annr(m)是环R的广义τ-本质右理想.称模M是广义τ-奇异(或广义τ-非奇异)的,如果其所有元素都是广义τ-奇异的(或唯一的广义τ-奇异元是0).讨论了模M的广义τ-奇异子模Zτ(M)的若干性质,给出了N■τM与M/N是广义τ-奇异的等价条件,证明了模M是广义τ-非奇异的,当且仅当对任意广义τ-奇异模N,Hom(N,M)=0.

In this paper,the concept of generalizedτ-(non)singularity is introduced.An element m in a module M is called generalizedτ-singular if its right annihilator is a generalizedτ-essential right ideal in R.The module M is generalizedτ-singular(resp.generalizedτ-nonsingular)if all its elemenets are generalizedτ-singular(resp.if the only generalizedτ-singular element is 0).Some properties of Zτ(M)which is a generalizedτ-singular submudules of M are discussed,the equivalent condition of N■τM and M/N being generalizedτ-singular is given,it is proved that a module M is generalizedτ-nonsingular if and only if Hom(N,M)=0 for all generalizedτ-singular modules N.