## $$(2, n)$$-ideals of commutative rings.(English)Zbl 1412.13005

Let $$R$$ be a commutative ring with $$1 \ne 0$$. In this paper, a new class of ideals called $$(2,n)$$-ideals is defined. An ideal $$I$$ is called $$(2,n)$$-ideal if whenever $$a,b,c\in R$$ and $$abc\in I$$, then $$ab \in I$$ or $$ac \in \sqrt{0}$$ or $$bc \in \sqrt{0}$$. Some properties of these ideals as well as the connection with prime ideals, primary ideals, $$2$$-absorbing primary ideals and $$n$$-ideals are studied.
In Theorem 2.6, the authors prove that a ring $$R$$ contains $$(2,n)$$-ideals if and only if $$R$$ has at most two minimal prime ideals. Using this result, in Theorem 2.12 they prove that $$R$$ is reduced with at most two minimal prime ideals if and only if $$(0)$$ is the only $$(2,n)$$-ideal of $$R$$.

### MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings

### Keywords:

primary ideals; 2-absorbing primary ideals; $$n$$-ideals
Full Text:

### References:

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