Pn –Ideal of Commutative Ring (JOURNAL OF UNIVERSITY OF BABYLON for Pure and Applied Sciences, 2023)
Background:This study gives a new generalization to Ids called -Id. If for all  with  and  then  and a proper Id P of  is known as a Id. It investigates some properties for example every element in  is nilpotent if  is an  of , of Pn-Ids analogous to n-Ids and PI. Some characterizations such as If  is a Id of   then  is also Id for generalization and it is proved that every element in - Ids is nilpotent. Accordingly, New versions of some theorems and proposition about Pn-Ids are given.Materials and Methods:In this paper we used the  ideal and  ideal to define ideal.Results:This strategy is continued in the second half of the study, when piecemeals are introduced as a generalization of  Id. A PI  of  is said to be a Id if the condition  with  implies  for all  The notion of  Id is given and some properties of Ids are investigated like to Ids. In Lemma 2.2, obtain every Id is Id. Also, if is a  iff   is an . It is proved (Proposition 2.5) that If  is a  Id in , then is a Id in  Conclusion:This study provides a new generalization to Ids called -Id. If for all  with  and  then  then a proper Id P of  is known as a Id. Some properties of Pn-Ids analogous to n-Ids and PI are investigated. Giving characterizations for such generalization proved that every element in  is nilpotent, when  is a Pn-Id. Consequently, new versions of some theorems and proposition about Pn-Ids are given.