Geodesic
On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture
Archivum mathematicum, Tome 52 (2016) no. 2, pp. 71-78 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \dots x_{n+1} \in I$ for $x_1, \ldots , x_{n+1} \in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that $\omega _{R[X]}(I[X])=\omega _R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $\omega _R(I)= \min \lbrace n\colon I \text{is} \text{an} n\text{-absorbing} \text{ideal} \text{of} R\rbrace $. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold: The ring $R$ is a Prüfer domain. The ring $R$ is a Gaussian ring such that its additive group is torsion-free. The additive group of the ring $R$ is torsion-free and $I$ is a radical ideal of $R$.
Let $R$ be a commutative ring with an identity different from zero and $n$ be a positive integer. Anderson and Badawi, in their paper on $n$-absorbing ideals, define a proper ideal $I$ of a commutative ring $R$ to be an $n$-absorbing ideal of $R$, if whenever $x_1 \dots x_{n+1} \in I$ for $x_1, \ldots , x_{n+1} \in R$, then there are $n$ of the $x_i$’s whose product is in $I$ and conjecture that $\omega _{R[X]}(I[X])=\omega _R(I)$ for any ideal $I$ of an arbitrary ring $R$, where $\omega _R(I)= \min \lbrace n\colon I \text{is} \text{an} n\text{-absorbing} \text{ideal} \text{of} R\rbrace $. In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following conditions hold: The ring $R$ is a Prüfer domain. The ring $R$ is a Gaussian ring such that its additive group is torsion-free. The additive group of the ring $R$ is torsion-free and $I$ is a radical ideal of $R$.
DOI : 10.5817/AM2016-2-71
Classification : 13A15, 13B02, 13B25, 13F05
Keywords: $n$-absorbing ideals; strongly $n$-absorbing ideals; polynomial rings; content algebras; Dedekind-Mertens content formula; Prüfer domains; Gaussian algebras; Gaussian rings 
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Nasehpour, Peyman. On the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture. Archivum mathematicum, Tome 52 (2016) no. 2, pp. 71-78. doi: 10.5817/AM2016-2-71

[1] Anderson, D.D., Camillo, V.: Armendariz rings and Gaussian rings. Comm. Algebra 26 (1998), 2265–2272. | DOI | MR | Zbl

[2] Anderson, D.D., Kang, B.G.: Content formulas for polynomials and power series and complete integral closure. J. Algebra 181 (1996), 82–94. | DOI | MR | Zbl

[3] Anderson, D.F., Badawi, A.: On $n$-absorbing ideals of commutative rings. Comm. Algebra 39 (2011), 1646–1672. | DOI | MR | Zbl

[4] Arnold, J.T., Gilmer, R.: On the content of polynomials. Proc. Amer. Math. Soc. 40 (1970), 556–562. | DOI | MR

[5] Badawi, A.: On $2$-absorbing ideals of commutative rings. Bull. Austral. Math. Soc. 75 (2007), 417–429. | DOI | MR | Zbl

[6] Bazzoni, S., Glaz, S.: Gaussian properties of total rings of quotients. J. Algebra 310 (1) (2007), 180–193. | DOI | MR | Zbl

[7] Bruns, W., Guerrieri, A.: The Dedekind-Mertens formula and determinantal rings. Proc. Amer. Math. Soc. 127 (3) (1999), 657–663. | DOI | MR | Zbl

[8] Darani, A.Y., Puczyłowski, E.R.: On $2$-absorbing commutative semigroups and their applications to rings. Semigroup Forum 86 (2013), 83–91. | DOI | MR | Zbl

[9] Eakin, P., Silver, J.: Rings which are almost polynomial rings. Trans. Amer. Math. Soc. 174 (1974), 425–449. | DOI | MR

[10] Epstein, N., Shapiro, J.: A Dedekind-Mertens theorem for power series rings. Proc. Amer. Math. Soc. 144 (2016), 917–924. | DOI | MR | Zbl

[11] Fields, D.E.: Zero divisors and nilpotent elements in power series rings. Proc. Amer. Math. Soc. 27 (3) (1971), 427–433. | DOI | MR | Zbl

[12] Gilmer, R.: Some applications of the Hilfssatz von Dedekind-Mertens. Math. Scand. 20 (1967), 240–244. | DOI | MR | Zbl

[13] Gilmer, R.: Multiplicative Ideal Theory. Marcel Dekker, New York, 1972. | MR | Zbl

[14] Gilmer, R., Grams, A., Parker, T.: Zero divisors in power series rings. J. Reine Angew. Math. 278 (1975), 145–164. | MR | Zbl

[15] Heinzer, W., Huneke, C.: The Dedekind-Mertens Lemma and the content of polynomials. Proc. Amer. Math. Soc. 126 (1998), 1305–1309. | DOI | MR

[16] Loper, K.A., Roitman, M.: The content of a Gaussian polynomial is invertible. Proc. Amer. Math. Soc. 133 (2005), 1267–1271. | DOI | MR | Zbl

[17] Nasehpour, P.: Zero-divisors of content algebras. Arch. Math. (Brno) 46 (4) (2010). | MR | Zbl

[18] Nasehpour, P.: Zero-divisors of semigroup modules. Kyungpook Math. J. 51 (1) (2011), 37–42. | DOI | MR | Zbl

[19] Nasehpour, P., Yassemi, S.: $M$-cancellation ideals. Kyungpook Math. J. 40 (2000), 259–263. | MR | Zbl

[20] Northcott, D.G.: A generalization of a theorem on the content of polynomials. Proc. Cambridge Philos. Soc. 55 (1959), 282–288. | MR | Zbl

[21] Ohm, J., Rush, D.E.: Content modules and algebras. Math. Scand. 39 (1972), 49–68. | DOI | MR | Zbl

[22] Prüfer, H.: Untersuchungen über Teilbarkeitseigenschaften in Körpern. J. Reine Angew. Math. 168 (1932), 1–36. | MR | Zbl

[23] Rege, M.B., Chhawchharia, S.: Armendariz rings. Proc. Japan Acad. Ser. A Math. Sci. 168 (1) (1997), 14–17. | MR | Zbl

[24] Rush, D.E.: Content algebras. Canad. Math. Bull. 21 (3) (1978), 329–334. | DOI | MR | Zbl

[25] Tsang, H.: Gauss’ Lemma. University of Chicago, Chicago, 1965, disseration. | MR | Zbl

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