Central quasipolar rings

In this paper, we introduce a kind of quasipolarity notion for rings, namely, an element a of a ring R is called central quasipolar if there exists p² = p ∈ R such that a + p is central in R, and the ring R is called central quasipolar if every element of R is central quasipolar. We give many characterizations and investigate general properties of central quasipolar rings. We determine the conditions that some subrings of upper triangular matrix rings are central quasipolar. A diagonal matrix over a local ring is characterized in terms of being central quasipolar. We prove that the class of central quasipolar rings lies between the classes of commutative rings and Dedekind nite rings, and a ring R is central quasipolar if and only if it is central clean. Further we show that several results of quasipolar rings can be extended to central quasipolar rings in this general setting.

DOI: https://doi.org/10.15446/recolma.v49n2.60446

Anillos casi-polares centrales

Mete B. Calci¹, Burcu Ungor¹, Abdullah Harmanci²

¹ Ankara University, Ankara, Turkey
e-mail: mburakcalci@gmail.com
e-mail: bungor@science.ankara.edu.tr
² Hacettepe University, Ankara, Turkey
e-mail: harmanci@hacettepe.edu.tr

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Abstract

In this paper, we introduce a kind of quasipolarity notion for rings, namely, an element a of a ring R is called central quasipolar if there exists p² = p ∈ R such that a+p is central in R, and the ring R is called central quasipolar if every element of R is central quasipolar. We give many characterizations and investigate general properties of central quasipolar rings. We determine the conditions that some subrings of upper triangular matrix rings are central quasipolar. A diagonal matrix over a local ring is characterized in terms of being central quasipolar. We prove that the class of central quasipolar rings lies between the classes of commutative rings and Dedekind finite rings, and a ring R is central quasipolar if and only if it is central clean. Further we show that several results of quasipolar rings can be extended to central quasipolar rings in this general setting.

Key words and phrases. Quasipolar ring, central quasipolar ring, clean ring, central clean ring.

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2010 Mathematics Subject Classification. 16S50, 16S70, 16U99.

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Resumen

En este trabajo, se presenta una noción de un tipo de casi-polaridad en anillos, esto es, un elemento a de un anillo R se dice casi-polar central si existe p² = p ∈ R tal que a + p es central en R, y el anillo R es llamado casi-polar central si todo elemento de R es casi-polar central. Se dan algunas caracterizaciones y se investigan propiedades generales de los anillos centrales casi-polares. Se determinan las condiciones bajo las cuales algunos subanillos de anillos de matrices triangulares superiores son casi-polares centrales. Una matriz diagonal sobre un anillo local se caracteriza en términos de ser casi-polar central. Se demuestra que la clase de anillos casi-polares centrales se encuentra dentro de la clase de los anillos conmutativos y los anillos finitos de Dedekind, y un anillo R es casi-polar central si es limpio central. Además se muestra que varios resultados de anillos casi-polares se pueden extender a anillos casi-polares centrales en un contexto general.

Palabras y frases clave. Anillo casi-polar, anillo casi-polar central, anillo limpio, anillo limpio central.

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Texto completo disponible en PDF

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References

[1] G. Borooah, A. J. Diesl, and T. J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (2008), no. 1, 281-296.

[2] H. Chen, Rings related to stable range conditions, World Scientific, 2011.

[3] J. Cui and J. Chen, When is a 2 × 2 matrix ring over a commutative local ring quasipolar?, Comm. Algebra 39 (2011), no. 9, 3212-3221.

[4] _______, A class of quasipolar rings, Comm. Algebra 40 (2012), no. 12, 4471-4482.

[5] A. J. Diesl, Nil clean rings, J. Algebra 383 (2013), 197-211.

[6] O. Gurgun, S. Halicioglu, and A. Harmanci, Quasipolar subrings of 3 × 3 matrix rings, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 21 (2013), no. 3, 133-146.

[7] _______, Nil-quasipolar rings, Bol. Soc. Mat. Mex. 20 (2014), 29-38.

[8] J. Han and W. K. Nicholson, Extensions of clean rings, Comm. Algebra 29 (2001), no. 6, 2589-2595.

[9] R. Harte, Invertibility and singularity for bounded linear operators, Marcel Dekker, 1988.

[10] D. Khurana, G. Marks, and A. Srivastava, On unit-central rings, Advances in ring theory, Trends Math., Birkhauser-Springer Basel AG, Basel (2010), 205-212.

[11] J. J. Koliha and P. Patricio, Elements of rings with equal spectral idempotents, J. Aust. Math. Soc. 72 (2002), no. 1, 137-152.

[12] W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278.

[13] _______, Strongly clean rings and fitting's lemma, Comm. Algebra 27 (1999), no. 8, 3583-3592.

[14] Z. Ying and J. Chen, On quasipolar rings, Algebra Colloq. 19 (2012), no. 4, 683-692.

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(Recibido en marzo de 2015. Aceptado en octubre de 2015)