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\((2, n)\)-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
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