扩张环上的$\Sigma$-相伴素理想

$\Sigma$-Associated Primes over Extension Rings

中文关键词:  相伴素理想  Ore扩张  $\Sigma$-相伴素理想

英文关键词:associated prime  Ore extension  $\Sigma$-associated prime

中文摘要:

      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.

英文摘要:

      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.