Semistar ascending chain conditions over polynomial rings

Abstract

Let D be an integral domain. G. Picozza associated to a stable semistar operation $$\star $$ on D, a semistar operation $$\star _1$$ on the polynomial ring D[X]. We defined the notion of semistar accr domain. We prove that D is $${\widetilde{\star }}$$ -Noetherian if and only D[X] is $$\star _1$$ -accr. On the other hand, we prove that D satisfies the ascending chain condition on radical quasi- $${\widetilde{\star }}$$ -ideals if and only if D[X] satisfies the ascending chain condition on radical quasi- $$\star _1$$ -ideals if and only if the Nagata ring of D with respect to the semistar operation $${\widetilde{\star }}$$ satisfies the ascending chain condition on radical ideals.