\((2, n)\)-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 \(R\).


13A15 Ideals and multiplicative ideal theory in commutative rings
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[1] Arnold, T., Krull dimension in power series rings, Trans. Amer. Math. Soc., 177, 299-304, (1973) · Zbl 0262.13007
[2] Badawi, A., On 2-absorbing ideals of commutative ring, Bull. Aust. Math. Soc., 75, 3, 417-429, (2007) · Zbl 1120.13004
[3] Badawi, A.; Tekir, U.; Yetkin, E., On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc., 51, 4, 1163-1173, (2014) · Zbl 1308.13001
[4] Badawi, A.; Tekir, U.; Yetkin, E., On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc., 52, 1, 97-111, (2015) · Zbl 1315.13008
[5] Badawi, A.; Tekir, U.; Ugurlu, E. A.; Ulucak, G.; Celikel, E. Y., Generalizations of 2-absorbing primary ideals of commutative rings, Turkish J. Math., 40, 3, 703-717, (2016) · Zbl 1424.13004
[6] Brewer, J., Power Series Over Commutative Rings, 64, (1981), Marcel Dekker Inc: Marcel Dekker Inc, New York · Zbl 0476.13015
[7] Calugareanu, G., \(U N\)-rings, J. Algebra Appl., 15, 10, 9, (2016)
[8] Hizem, S.; Benhissi, A., Nonnil-Noetherian rings and the \(S F T\) property, Rocky Mountain J. Math., 41, 5, 1483-1500, (2011) · Zbl 1242.13027
[9] Malek, A.; Hamed, A.; Benhissi, A., 2-absorbing ideals in formal power series rings, Palest. J. Math., 6, 2, 502-506, (2017) · Zbl 1371.13003
[10] Tekir, U.; Koc, S.; Oral, K. H., \(n\)-Ideals of commutative rings, Filomat, 31, 10, 2933-2941, (2017)
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