Skew PBW Extensions of Baer, quasi-Baer,
p.p. and p.q.-rings

ARMANDO REYES^*

Universidad Nacional de Colombia, Departamento de Matemáticas, Bogotá, Colombia.

Abstract. The aim of this paper is to study skew Poincaré-Birkhoff-Witt extensions of Baer, quasi-Baer, p.p. and p.q.-Baer rings. Using a notion of rigidness, we prove that these properties are stable over this kind of extensions.

Keywords: Baer, quasi-Baer, p.p. and p.q.-Baer rings, skew Poincaré- Birkhoff-Witt extensions.
MSC2010: 16E50, 16S36, 16D25.

Extensiones PBW torcidas de anillos de Baer,
quasi-Baer, p.p. y p.q-Baer

Resumen. El propósito de este artículo es estudiar las extensiones torcidas de Poincaré-Birkhoff-Witt de anillos de Baer, quasi-Baer, p.p. y p.q.-Baer. Utilizando una noción de rigidez, probamos que estas propiedades son estables para esta clase de extensiones.

Palabras clave: Anillos Baer, quasi-Baer, p.p, p.q.-Baer, extensiones torcidas de Poincaré-Birkhoff-Witt.

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Referencias

[1] Armendariz E.P., "A note on extensions of Baer and p.p.-rings", J. Austral. Math. Soc. 18 (1974), 470-473.

[2] Armendariz E.P., Koo H.K. and Park J.K., "Isomorphic Ore extensions", Comm. Algebra 15 (1987), No. 12, 2633-2652.

[3] Birkenmeier G.F., "Baer rings and quasicontinuous rings have a MDSN", Pacific J. Math. 97 (1981), No. 2, 283-292.

[4] Birkenmeier G.F., Kim J.Y. and Park J.K., "Principally quasi-Baer rings", Comm. Algebra 29 (2001), No. 2, 639-660.

[5] Birkenmeier G.F., Kim J.Y. and Park J.K., "Polynomial extensions of Baer and quasi-Baer rings", J. Pure Appl. Algebra 159 (2001), No. 1, 25-42.

[6] Clark W.E., "Twisted matrix units semigroup algebras", Duke Math. J. 34 (1967), 417-423.

[7] Gallego C. and Lezama O., "Gröbner bases for ideals of σ-PBW extensions", Comm. Algebra 39 (2011), No. 1, 50-75.

[8] Han J., Hirano Y. and Kim H., "Semiprime Ore extensions", Comm. Algebra 28 (2000), No. 8, 3795-3801.

[9] Han J., Hirano Y. and Kim H., "Some results on skew polynomial rings over a reduced ring", in International Symposium on Ring Theory (Kyongju, 1999), Trends Math., Birkhäuser Boston, Boston, MA (2001), 123-129.

[10] Hashemi E., Moussavi A. and Seyyed Javadi H.H., "Polynomial Ore extensions of Baer and p.p.-rings", Bull. Iranian Math. Soc. 29 (2003), No. 2, 65-86.

[11] Hashemi E. and Moussavi A., "Polynomial extensions of quasi-Baer rings", Acta Math. Hungar. 107 (2005), No. 3, 207-224.

[12] Hinchcliffe O., "Difussion Algebras", Thesis (Ph.D.), University of Sheffield, Sheffield, 2005, 118 p.

[13] Hong C.Y., Kim N.K. and Kwak T.K., "Ore extensions of Baer and p.p.-rings", J. Pure Appl. Algebra 151 (2000), No. 3, 215-226.

[14] Hong C.Y., Kim N.K. and Lee Y., "Ore extensions of quasi-Baer rings", Comm. Algebra 37 (2009), No. 6, 2030-2039.

[15] Kaplansky I., Rings of Operators. W.A. Benjamin, Inc., New York-Amsterdam, 1968.

[16] Krempa J., "Some Examples of reduced rings", Algebra Colloq. 3 (1996), No. 4, 289-300.

[17] Lezama O. and Reyes A., "Some homological properties of skew PBW extensions", Comm. Algebra 42 (2014), No. 3, 1200-1230.

[18] Lezama O., Acosta J.P. and Reyes A., "Prime ideals of skew PBW extensions", Rev. Un. Mat. Argentina 56 (2015), No. 2, 39-55.

[19] Matczuk J., "A Characterization of σ-rigid rings", Comm. Algebra 32 (2004), No. 11, 4333- 4336.

[20] McConnell J.C. and Robson J.C., Noncommutative Noetherian rings, Graduate Studies in Mathematics, 30, American Mathematical Society, Providence, RI, 2001.

[21] Nasr-Isfahani A.R. and Moussavi A., "Baer and quasi-Baer differential polynomial rings", Comm. Algebra 36 (2008), No. 9, 3533-3542.

[22] Passman D.S., "Prime ideals in enveloping rings", Trans. Amer. Math. Soc. 302 (1987), No. 2, 535-560.

[23] Reyes A., "Ring and module theoretical properties of skew PBW extensions", Thesis (Ph.D.), Universidad Nacional de Colombia, Bogotá, 2013, 142 p.

[24] Reyes A., "Jacobson's conjecture and skew PBW extensions", Rev. Integr. Temas Mat. 32 (2014), No. 2, 139-152.

^*E-mail: mareyesv@unal.edu.co
Received: 25 September 2015, Accepted: 17 November 2015.
To cite this article: A. Reyes, Skew PBW Extensions of Bear, quasi-Baer, p.p. and p.q.-rings, Rev. Integr. Temas Mat. 33 (2015), No. 2, 173-189.