Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
$n$-absorbing ideals; strongly $n$-absorbing ideals; polynomial rings; content algebras; Dedekind-Mertens content formula; Prüfer domains; Gaussian algebras; Gaussian rings
$n$-absorbing ideals; strongly $n$-absorbing ideals; polynomial rings; content algebras; Dedekind-Mertens content formula; Prüfer domains; Gaussian algebras; Gaussian rings
Summary:
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 $R$.
References:
[1] Anderson, D.D., Camillo, V.: Armendariz rings and Gaussian rings. Comm. Algebra 26 (1998), 2265–2272. DOI 10.1080/00927879808826274 | MR 1626606 | Zbl 0915.13001
[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 10.1006/jabr.1996.0110 | MR 1382027 | Zbl 0857.13017
[3] Anderson, D.F., Badawi, A.: On $n$-absorbing ideals of commutative rings. Comm. Algebra 39 (2011), 1646–1672. DOI 10.1080/00927871003738998 | MR 2821499 | Zbl 1232.13001
[4] Arnold, J.T., Gilmer, R.: On the content of polynomials. Proc. Amer. Math. Soc. 40 (1970), 556–562. DOI 10.1090/S0002-9939-1970-0252360-3 | MR 0252360
[5] Badawi, A.: On $2$-absorbing ideals of commutative rings. Bull. Austral. Math. Soc. 75 (2007), 417–429. DOI 10.1017/S0004972700039344 | MR 2331019 | Zbl 1120.13004
[6] Bazzoni, S., Glaz, S.: Gaussian properties of total rings of quotients. J. Algebra 310 (1) (2007), 180–193. DOI 10.1016/j.jalgebra.2007.01.004 | MR 2307788 | Zbl 1118.13020
[7] Bruns, W., Guerrieri, A.: The Dedekind-Mertens formula and determinantal rings. Proc. Amer. Math. Soc. 127 (3) (1999), 657–663. DOI 10.1090/S0002-9939-99-04535-9 | MR 1468185 | Zbl 0915.13008
[8] Darani, A.Y., Puczyłowski, E.R.: On $2$-absorbing commutative semigroups and their applications to rings. Semigroup Forum 86 (2013), 83–91. DOI 10.1007/s00233-012-9417-z | MR 3016263 | Zbl 1270.20064
[9] Eakin, P., Silver, J.: Rings which are almost polynomial rings. Trans. Amer. Math. Soc. 174 (1974), 425–449. DOI 10.1090/S0002-9947-1972-0309924-4 | MR 0309924
[10] Epstein, N., Shapiro, J.: A Dedekind-Mertens theorem for power series rings. Proc. Amer. Math. Soc. 144 (2016), 917–924. DOI 10.1090/proc/12661 | MR 3447645 | Zbl 1332.13018
[11] Fields, D.E.: Zero divisors and nilpotent elements in power series rings. Proc. Amer. Math. Soc. 27 (3) (1971), 427–433. DOI 10.1090/S0002-9939-1971-0271100-6 | MR 0271100 | Zbl 0219.13023
[12] Gilmer, R.: Some applications of the Hilfssatz von Dedekind-Mertens. Math. Scand. 20 (1967), 240–244. DOI 10.7146/math.scand.a-10833 | MR 0236159 | Zbl 0167.03602
[13] Gilmer, R.: Multiplicative Ideal Theory. Marcel Dekker, New York, 1972. MR 0427289 | Zbl 0248.13001
[14] Gilmer, R., Grams, A., Parker, T.: Zero divisors in power series rings. J. Reine Angew. Math. 278 (1975), 145–164. MR 0387274 | Zbl 0309.13009
[15] Heinzer, W., Huneke, C.: The Dedekind-Mertens Lemma and the content of polynomials. Proc. Amer. Math. Soc. 126 (1998), 1305–1309. DOI 10.1090/S0002-9939-98-04165-3 | MR 1425124
[16] Loper, K.A., Roitman, M.: The content of a Gaussian polynomial is invertible. Proc. Amer. Math. Soc. 133 (2005), 1267–1271. DOI 10.1090/S0002-9939-04-07826-8 | MR 2111931 | Zbl 1137.13301
[17] Nasehpour, P.: Zero-divisors of content algebras. Arch. Math. (Brno) 46 (4) (2010). MR 2754063 | Zbl 1240.13002
[18] Nasehpour, P.: Zero-divisors of semigroup modules. Kyungpook Math. J. 51 (1) (2011), 37–42. DOI 10.5666/KMJ.2011.51.1.037 | MR 2784646 | Zbl 1218.13005
[19] Nasehpour, P., Yassemi, S.: $M$-cancellation ideals. Kyungpook Math. J. 40 (2000), 259–263. MR 1803117 | Zbl 1020.13002
[20] Northcott, D.G.: A generalization of a theorem on the content of polynomials. Proc. Cambridge Philos. Soc. 55 (1959), 282–288. MR 0110732 | Zbl 0103.27102
[21] Ohm, J., Rush, D.E.: Content modules and algebras. Math. Scand. 39 (1972), 49–68. DOI 10.7146/math.scand.a-11411 | MR 0344289 | Zbl 0248.13013
[22] Prüfer, H.: Untersuchungen über Teilbarkeitseigenschaften in Körpern. J. Reine Angew. Math. 168 (1932), 1–36. MR 1581355 | Zbl 0004.34001
[23] Rege, M.B., Chhawchharia, S.: Armendariz rings. Proc. Japan Acad. Ser. A Math. Sci. 168 (1) (1997), 14–17. MR 1442245 | Zbl 0960.16038
[24] Rush, D.E.: Content algebras. Canad. Math. Bull. 21 (3) (1978), 329–334. DOI 10.4153/CMB-1978-057-8 | MR 0511581 | Zbl 0441.13005
[25] Tsang, H.: Gauss’ Lemma. University of Chicago, Chicago, 1965, disseration. MR 2611536 | Zbl 0266.13007
Search
Partner of
