Research Article
BibTex RIS Cite
Year 2024, Early Access, 1 - 15
https://doi.org/10.24330/ieja.1578725

Abstract

References

  • R. F. Arens and I. Kaplansky, Topological representation of algebras, Trans. Amer. Math. Soc., 63 (1948), 457-481.
  • G. Azumaya, Strongly $\pi$-regular rings, J. Fac. Sci. Hokkaido Univ. Ser. I, 13 (1954), 34-39.
  • G. M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29(2) (1978), 178-218.
  • W. C. Brown, Matrices Over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, 169, Marcel Dekker, Inc., New York, 1993.
  • G. Calugareanu, Tripotents: a class of strongly clean elements in rings, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat., 26(1) (2018), 69-80.
  • G. Calugareanu, The formula $ABA=Tr(AB)A$ for matrices, Beitr. Algebra Geom., (2024), https://doi.org/10.1007/s13366-024-00756-9.
  • G. Ehrlich, Unit-regular rings, Portugal. Math., 27 (1968), 209-212.
  • W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra, 27(8) (1999), 3583-3592.

On the equation $a^{2}x=a$ in unital rings

Year 2024, Early Access, 1 - 15
https://doi.org/10.24330/ieja.1578725

Abstract

In an arbitrary ring, the equation in the title defines the left strongly regular elements. Elements that are both left and right strongly regular are simply called strongly regular. If all elements of a ring are left strongly regular, then they are, in fact, strongly regular and this is the definition of strongly regular rings.

We provide a characterization of when a left strongly regular element is indeed strongly regular, based on an intrinsic condition. While we show that it is not possible to give $2\times 2$ non-examples over $\mathbb{Z}$ or in certain matrix rings over $\mathbb{Z}_{n}$ for $n\in \{8,9,16\}$, we present two examples by George Bergman: a left strongly regular element that is not regular and a regular, left strongly regular element that is not strongly regular.

Further, we prove results for left strongly regular (square) matrices over various types of rings and propose a conjecture, strongly supported by computational evidence: over commutative rings, left strongly regular matrices are strongly regular.

References

  • R. F. Arens and I. Kaplansky, Topological representation of algebras, Trans. Amer. Math. Soc., 63 (1948), 457-481.
  • G. Azumaya, Strongly $\pi$-regular rings, J. Fac. Sci. Hokkaido Univ. Ser. I, 13 (1954), 34-39.
  • G. M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29(2) (1978), 178-218.
  • W. C. Brown, Matrices Over Commutative Rings, Monographs and Textbooks in Pure and Applied Mathematics, 169, Marcel Dekker, Inc., New York, 1993.
  • G. Calugareanu, Tripotents: a class of strongly clean elements in rings, An. Stiint. Univ. "Ovidius" Constanta Ser. Mat., 26(1) (2018), 69-80.
  • G. Calugareanu, The formula $ABA=Tr(AB)A$ for matrices, Beitr. Algebra Geom., (2024), https://doi.org/10.1007/s13366-024-00756-9.
  • G. Ehrlich, Unit-regular rings, Portugal. Math., 27 (1968), 209-212.
  • W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra, 27(8) (1999), 3583-3592.
There are 8 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Grigore Călugăreanu

Horia F. Pop This is me

Early Pub Date November 4, 2024
Publication Date
Submission Date June 22, 2024
Acceptance Date September 24, 2024
Published in Issue Year 2024 Early Access

Cite

APA Călugăreanu, G., & Pop, H. F. (2024). On the equation $a^{2}x=a$ in unital rings. International Electronic Journal of Algebra1-15. https://doi.org/10.24330/ieja.1578725
AMA Călugăreanu G, Pop HF. On the equation $a^{2}x=a$ in unital rings. IEJA. Published online November 1, 2024:1-15. doi:10.24330/ieja.1578725
Chicago Călugăreanu, Grigore, and Horia F. Pop. “On the Equation $a^{2}x=a$ in Unital Rings”. International Electronic Journal of Algebra, November (November 2024), 1-15. https://doi.org/10.24330/ieja.1578725.
EndNote Călugăreanu G, Pop HF (November 1, 2024) On the equation $a^{2}x=a$ in unital rings. International Electronic Journal of Algebra 1–15.
IEEE G. Călugăreanu and H. F. Pop, “On the equation $a^{2}x=a$ in unital rings”, IEJA, pp. 1–15, November 2024, doi: 10.24330/ieja.1578725.
ISNAD Călugăreanu, Grigore - Pop, Horia F. “On the Equation $a^{2}x=a$ in Unital Rings”. International Electronic Journal of Algebra. November 2024. 1-15. https://doi.org/10.24330/ieja.1578725.
JAMA Călugăreanu G, Pop HF. On the equation $a^{2}x=a$ in unital rings. IEJA. 2024;:1–15.
MLA Călugăreanu, Grigore and Horia F. Pop. “On the Equation $a^{2}x=a$ in Unital Rings”. International Electronic Journal of Algebra, 2024, pp. 1-15, doi:10.24330/ieja.1578725.
Vancouver Călugăreanu G, Pop HF. On the equation $a^{2}x=a$ in unital rings. IEJA. 2024:1-15.