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On skew polynomials over Ikeda-Nakayama rings. (English) Zbl 1501.16020

Summary: A ring \(R\) is called a left Ikeda-Nakayama ring (left IN-ring) if the right annihilator of the intersection of any two left ideals is the sum of the two right annihilators. Also a ring \(R\) is called a right SA-ring if the sum of right annihilators of two ideals is a right annihilator of an ideal of \(R\). In this paper for a compatible endomorphism \(\alpha\) of \(R\), we show that: (i) If \(R[x;\alpha]\) is a left IN-ring, then \(R\) is an Armendariz left IN-ring. (ii) If \(R\) is a reduced left IN-ring with finitely many minimal prime ideals, then \(R[x;\alpha]\) is a left IN-ring. (iii) \(R[x;\alpha]\) is a right SA-ring, if and only if \(R\) is a quasi-Armendariz right SA-ring. We give a class of non-reduced rings \(R\) such that \(R[x]\) is left IN-ring. Also we give some examples to show that assumption compatibility on \(\alpha\) is not superfluous.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16D25 Ideals in associative algebras
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[1] Baser, M.; Kwa, T. K., Quasi-Armendariz property for skew polynomial rings, Comm. Korean Math. Soc, 26, 4, 557-573 (2011) · Zbl 1236.16025 · doi:10.4134/CKMS.2011.26.4.557
[2] Birkenmeier, G. F.; Ghirati, M.; Taherifar, A., When is a sum of annihilator ideals an annihilator ideal?, Commun. Algebra, 43, 7, 2690-2702 (2015) · Zbl 1355.16004 · doi:10.1080/00927872.2014.882931
[3] Brown, K. A.; Goodearl, K. R., Lectures on Algebraic Quantum Groups (2002), Basel: Birkhäuser · Zbl 1027.17010
[4] Camillo, V.; Nicholson, W. K.; Yousif, M. F., Ikeda-Nakayama rings, J. Algebra, 226, 2, 1001-1010 (2000) · Zbl 0958.16002 · doi:10.1006/jabr.1999.8217
[5] Cortes, W., Algebraic Structures and Their Representations. Contemporary Mathematics, 376, Skew Armendariz rings and annihilator ideals of skew polynomial rings, 249-259 (2005), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1080.16019
[6] Goodearl, K. R.; Warfield, R. B. Jr., An Introduction to Noncommutative Noetherian Rings (2004), Cambridge: Cambridge University Press · Zbl 1101.16001
[7] Hajarnavis, C. R.; Norton, N. C., On dual rings and their modules, J. Algebra, 93, 2, 253-266 (1985) · Zbl 0595.16009 · doi:10.1016/0021-8693(85)90159-0
[8] Han, J.; Hirano, Y.; Kim, H., Semiprime Ore extensions, Commun. Algebra, 28, 8, 3795-3801 (2000) · Zbl 0965.16015 · doi:10.1080/00927870008827058
[9] Hashemi, E., Compatible ideals and radicals of Ore extensions, New York J. Math, 12, 349-356 (2006) · Zbl 1106.16029
[10] Hashemi, E., Prime ideals and strongly prime ideals of skew Laurent polynomial rings, Int. J. Math. Math. Sci, 835605 (2008) · Zbl 1165.16024
[11] Hashemi, E.; Hamidizadeh, M.; Alhevaz, A., Some types of ring elements in Ore extensions over noncommutative rings, J. Algebra Appl, 16, 11, 1750201 (2017) · Zbl 1392.16034 · doi:10.1142/S0219498817502012
[12] Hashemi, E.; Moussavi, A., Polynomial extensions of quasi-Baer rings, Acta Math. Hung, 107, 3, 207-224 (2005) · Zbl 1081.16032 · doi:10.1007/s10474-005-0191-1
[13] Hirano, Y., On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168, 1, 45-52 (2002) · Zbl 1007.16020 · doi:10.1016/S0022-4049(01)00053-6
[14] Hong, C. Y.; Kim, N. K.; Kwak, T. K., On skew Armendariz rings, Commun. Algebra, 31, 1, 103-122 (2003) · Zbl 1042.16014 · doi:10.1081/AGB-120016752
[15] Hong, C. Y.; Kim, N. K.; Lee, Y., Skew polynomial rings over semiprime rings, J. Korean Math. Soc, 47, 5, 879-897 (2010) · Zbl 1207.16028 · doi:10.4134/JKMS.2010.47.5.879
[16] Ikeda, M.; Nakayama, T., On some characteristic properties of quasi-Frobenius and regular rings, Proc. Am. Math. Soc, 5, 1, 15-19 (1954) · Zbl 0055.02602 · doi:10.1090/S0002-9939-1954-0060489-9
[17] Kaplansky, I., Dual rings, Ann. Math, 49, 3, 689-701 (1948) · Zbl 0031.34401 · doi:10.2307/1969052
[18] Kosan, T. M., The Armendariz module and its application to the Ikeda-Nakayama module, Int. J. Math. Math. Sci, 2006, 1-7 (2006) · Zbl 1143.16008
[19] Leroy, A.; Matczuk, J., Goldie conditions for ore extensions over semiprime rings, Algebr. Represent. Theor, 8, 5, 679-688 (2005) · Zbl 1090.16011 · doi:10.1007/s10468-005-0707-y
[20] Lee, T. K.; Zhou, Y., Armendariz and reduced rings, Commun. Algebra, 32, 6, 2287-2299 (2004) · Zbl 1068.16037 · doi:10.1081/AGB-120037221
[21] McConnell, J. C.; Robson, J. C., Noncommutative Noetherian Rings (2001), Chichester: John Wiley & Sons, Chichester · Zbl 0980.16019
[22] Mohammadi, R.; Moussavi, A.; Zahiri, M., A note on minimal prime ideals, Bull. Korean Math. Soc, 54, 4, 1281-1291 (2017) · Zbl 1381.16003
[23] Rege, M. B.; Chhawchharia, S., Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci, 73, 1, 14-17 (1997) · Zbl 0960.16038 · doi:10.3792/pjaa.73.14
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