(untitled) (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.