Invo-regular unital rings

Peter V. Danchev

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2018)

  • Volume: 72, Issue: 1
  • ISSN: 0365-1029

Abstract

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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.

How to cite

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Peter V. Danchev. "Invo-regular unital rings." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 72.1 (2018): null. <http://eudml.org/doc/290767>.

@article{PeterV2018,
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.},
author = {Peter V. Danchev},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Unit-regular rings; clean rings; strongly clean rings; idempotents; involutions; nilpotents; units},
language = {eng},
number = {1},
pages = {null},
title = {Invo-regular unital rings},
url = {http://eudml.org/doc/290767},
volume = {72},
year = {2018},
}

TY - JOUR
AU - Peter V. Danchev
TI - Invo-regular unital rings
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2018
VL - 72
IS - 1
SP - null
AB - 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.
LA - eng
KW - Unit-regular rings; clean rings; strongly clean rings; idempotents; involutions; nilpotents; units
UR - http://eudml.org/doc/290767
ER -

References

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  1. Camillo, V. P., Khurana, D., A characterization of unit regular rings, Commun. Algebra 29 (2001), 2293-2295. 
  2. Danchev, P. V., A new characterization of Boolean rings with identity, Irish Math. Soc. Bull. 76 (2015), 55-60. 
  3. Danchev, P. V., On weakly clean and weakly exchange rings having the strong property, Publ. Inst. Math. Beograd 101 (2017), 135-142. 
  4. Danchev, P. V., Invo-clean unital rings, Commun. Korean Math. Soc. 32 (2017), 19-27. 
  5. Danchev, P. V., Lam, T. Y., Rings with unipotent units, Publ. Math. Debrecen 88 (2016), 449-466. 
  6. Ehrlich, G., Unit-regular rings, Portugal. Math. 27 (1968), 209-212. 
  7. Goodearl, K. R., Von Neumann Regular Rings, Second Edition, Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. 
  8. Hartwig, R. E., Luh, J., A note on the group structure of unit regular ring elements, Pacific J. Math. 71 (1977), 449-461. 
  9. Hirano, Y., Tominaga, H., Rings in which every element is the sum of two idempotents, Bull. Austral. Math. Soc. 37 (1988), 161-164. 
  10. Lam, T. Y., A First Course in Noncommutative Rings, Second Edition, Springer-Verlag, Berlin-Heidelberg-New York, 2001. 
  11. Lam, T. Y., Murray, W., Unit regular elements in corner rings, Bull. Hong Kong Math. Soc. 1 (1997), 61-65. 
  12. Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269-278. 
  13. Nicholson, W. K., Strongly clean rings and Fitting’s lemma, Commun. Algebra 27 (1999), 3583-3592. 
  14. 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. 

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