## 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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### References:

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