$\Sigma$-Associated Primes over Extension Rings

DOI：10.3770/j.issn:2095-2651.2015.05.004

 作者 单位 欧阳伦群 湖南科技大学数学与计算科学学院, 湖南 湘潭 411201 刘金旺 湖南科技大学数学与计算科学学院, 湖南 湘潭 411201 向跃明 怀化学院数学与应用数学系, 湖南 怀化 418000

作为对相伴素理想与幂零相伴素理想的推广,我们在本文中引进了$\Sigma$-相伴素理想的定义,探讨了$\Sigma$-相伴素理想的基本性质,证明了Ore扩张环$R[x;\alpha,\delta]$、斜洛朗多项式环$R[x,x^{-1};\alpha]$及斜幂级数环$R[[x;\alpha]]$的$\Sigma$-相伴素理想都分别可以用环$R$的$\Sigma$-相伴素理想来刻画,从而将相伴素理想与幂零相伴素理想的一些已有结论推广到更一般的情形.

In this paper we introduce a concept, called $\Sigma$-associated primes, that is a generalization of both associated primes and nilpotent associated primes. We first observe the basic properties of $\Sigma$-associated primes and construct typical examples. We next describe all $\Sigma$-associated primes of the Ore extension $R[x;\alpha,\delta]$, the skew Laurent polynomial ring $R[x,x^{-1};\alpha]$ and the skew power series ring $R[[x;\alpha]]$, in terms of the $\Sigma$-associated primes of $R$ in a very straightforward way. Consequently several known results relating to associated primes and nilpotent associated primes are extended to a more general setting.