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On generalizations of prime ideals. (English) Zbl 1278.13003
Summary: Let \(R\) be a commutative ring with identity. Let \(\phi: S(R)\to S(R)\cup\{\varnothing\}\) be a function, where \(S(R)\) is the set of ideals of \(R\). Suppose \(n\geq 2\) is a positive integer. A nonzero proper ideal \(I\) of \(R\) is called \((n- 1,n)-\phi\)-prime if, whenever \(a_1, a_2,\dots, a_n\in R\) and \(a_1, a_2,\dots,a_n\in I\setminus\phi(I)\), the product of \((n-1)\) of the \(a_i\)’s is in \(I\).
In this article, we study \((n- 1,n)-\phi\)-prime ideals \((n\geq 2)\). A number of results concerning \((n- 1,n)-\phi\)-prime ideals and examples of \((n- 1,n)-\phi\)-prime ideals are also given. Finally, rings with the property that for some \(\phi\), every proper ideal is \((n- 1,n)-\phi\)-prime, are characterized.

MSC:
13A15 Ideals and multiplicative ideal theory in commutative rings
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