Nonnil-Laskerian rings

Let R be a commutative ring with unity. In this paper we introduce the concept of Nonnil-Laskerian ring that is related to the class of Laskerian rings. A ring R is said to be Nonnil-Laskerian if every nonnil ideal I of R is decomposable. We show that Nonnil-Laskerian rings enjoy analogs of many properties of Laskerian ring. We give an example of Nonnil-Laskerian ring, wich is not Laskerian. We study the Nonnil-Laskerian property over the polynomial and formel power series rings. In particular, we show that we have not an equivalence between Nonnil-Laskerian and Nonnil-Noetherian concepts in R[[X]] and R[X], contrary to the Laskerian and Noetherian concepts.