Invo-regular unital rings

Peter V. Danchev

Abstract


It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.

Keywords


Unit-regular rings; clean rings; strongly clean rings; idempotents; involutions; nilpotents; units

Full Text:

PDF

References


Camillo, V. P., Khurana, D., A characterization of unit regular rings, Commun. Algebra 29 (2001), 2293-2295.

Danchev, P. V., A new characterization of Boolean rings with identity, Irish Math. Soc. Bull. 76 (2015), 55-60.

Danchev, P. V., On weakly clean and weakly exchange rings having the strong property, Publ. Inst. Math. Beograd 101 (2017), 135-142.

Danchev, P. V., Invo-clean unital rings, Commun. Korean Math. Soc. 32 (2017), 19-27.

Danchev, P. V., Lam, T. Y., Rings with unipotent units, Publ. Math. Debrecen 88 (2016), 449-466.

Ehrlich, G., Unit-regular rings, Portugal. Math. 27 (1968), 209-212.

Goodearl, K. R., Von Neumann Regular Rings, Second Edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991.

Hartwig, R. E., Luh, J., A note on the group structure of unit regular ring elements, Pacific J. Math. 71 (1977), 449-461.

Hirano, Y., Tominaga, H., Rings in which every element is the sum of two idempotents, Bull. Austral. Math. Soc. 37 (1988), 161-164.

Lam, T. Y., A First Course in Noncommutative Rings, Second Edition, Springer-Verlag, Berlin-Heidelberg-New York, 2001.

Lam, T. Y., Murray, W., Unit regular elements in corner rings, Bull. Hong Kong Math. Soc. 1 (1997), 61-65.

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

Nicholson, W. K., Strongly clean rings and Fitting’s lemma, Commun. Algebra 27 (1999), 3583-3592.

Nielsen, P. P., Ster, J., Connections between unit-regularity, regularity, cleanness and strong cleanness of elements and rings, Trans. Amer. Math. Soc. 370 (2018), 1759-1782.




DOI: http://dx.doi.org/10.17951/a.2018.72.1.45-53
Date of publication: 2018-06-25 09:04:05
Date of submission: 2018-06-24 22:15:17


Statistics


Total abstract view - 830
Downloads (from 2020-06-17) - PDF - 457

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2018 Peter V. Danchev