$\Sigma$-Associated Primes over Extension Rings

Abstract:

      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.