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\(I\)-prime ideals. (English) Zbl 1355.13003

Summary: In this paper, we introduce a new generalization of weakly prime ideals called \(I\)-prime. Suppose \(R\) is a commutative ring with identity and \(I\) a fixed ideal of \(R\). A proper ideal \(P\) of \(R\) is \(I\)-prime if for \(a,b\in R\) with \(ab\in P-IP\) implies either \(a\in P\) or \(b\in P\). We give some characterizations of \(I\)-prime ideals and study some of its properties. Moreover, we give conditions under which \(I\)-prime ideals becomes prime or weakly prime and we construct the view of \(I\)-prime ideal in decomposite rings.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
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