Nil-clean and unit-regular elements in certain subrings of 𝕄 2 ( ℤ )

Yansheng Wu; Gaohua Tang; Guixin Deng; Yiqiang Zhou

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 1, page 197-205
  • ISSN: 0011-4642

Abstract

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An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of 𝕄 2 ( ℤ ) .

How to cite

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Wu, Yansheng, et al. "Nil-clean and unit-regular elements in certain subrings of ${\mathbb {M}}_2(\mathbb {Z})$." Czechoslovak Mathematical Journal 69.1 (2019): 197-205. <http://eudml.org/doc/294515>.

@article{Wu2019,
abstract = {An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of $\mathbb \{M\}_2(\mathbb \{Z\})$.},
author = {Wu, Yansheng, Tang, Gaohua, Deng, Guixin, Zhou, Yiqiang},
journal = {Czechoslovak Mathematical Journal},
keywords = {clean element; nil-clean element; unit-regular element; Jacobson's lemma for nil-clean elements},
language = {eng},
number = {1},
pages = {197-205},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nil-clean and unit-regular elements in certain subrings of $\{\mathbb \{M\}\}_2(\mathbb \{Z\})$},
url = {http://eudml.org/doc/294515},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Wu, Yansheng
AU - Tang, Gaohua
AU - Deng, Guixin
AU - Zhou, Yiqiang
TI - Nil-clean and unit-regular elements in certain subrings of ${\mathbb {M}}_2(\mathbb {Z})$
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 197
EP - 205
AB - An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements that are not clean in a ring. The rings under consideration in our examples are particular subrings of $\mathbb {M}_2(\mathbb {Z})$.
LA - eng
KW - clean element; nil-clean element; unit-regular element; Jacobson's lemma for nil-clean elements
UR - http://eudml.org/doc/294515
ER -

References

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  1. Andrica, D., Călugăreanu, G., 10.1142/S0219498814500091, J. Algebra Appl. 13 (2014), Article ID 1450009, 9 pages. (2014) Zbl1294.16019MR3195166DOI10.1142/S0219498814500091
  2. Camillo, V. P., Khurana, D., 10.1081/AGB-100002185, Commun. Algebra 29 (2001), 2293-2295. (2001) Zbl0992.16011MR1837978DOI10.1081/AGB-100002185
  3. Chen, J., Yang, X., Zhou, Y., 10.1080/00927870600860791, Commun. Algebra 34 (2006), 3659-3674. (2006) Zbl1114.16024MR2262376DOI10.1080/00927870600860791
  4. Cvetkovic-Ilic, D., Harte, R., 10.1016/j.aml.2009.11.009, Appl. Math. Lett. 23 (2010), 417-420. (2010) Zbl1195.16033MR2594854DOI10.1016/j.aml.2009.11.009
  5. Diesl, A. J., 10.1016/j.jalgebra.2013.02.020, J. Algebra 383 (2013), 197-211. (2013) Zbl1296.16016MR3037975DOI10.1016/j.jalgebra.2013.02.020
  6. Khurana, D., Lam, T. Y., 10.1016/j.jalgebra.2004.04.019, J. Algebra 280 (2004), 683-698. (2004) Zbl1067.16050MR2090058DOI10.1016/j.jalgebra.2004.04.019
  7. Koşan, T., Wang, Z., Zhou, Y., 10.1016/j.jpaa.2015.07.009, J. Pure Appl. Algebra 220 (2016), 633-646. (2016) Zbl1335.16026MR3399382DOI10.1016/j.jpaa.2015.07.009
  8. Lam, T. Y., Nielsen, P. P., 10.1090/conm/609/12127, Ring Theory and Its Applications D. V. Huynh et al. Contemporary Mathematics 609, American Mathematical Society, Providence (2014), 185-195. (2014) Zbl1294.15005MR3204360DOI10.1090/conm/609/12127
  9. Tang, G., Zhou, Y., 10.1016/j.laa.2013.02.019, Linear Algebra Appl. 438 (2013), 4672-4688. (2013) Zbl1283.16026MR3039217DOI10.1016/j.laa.2013.02.019
  10. Zhuang, G., Chen, J., Cui, J., 10.1016/j.laa.2011.07.044, Linear Algebra Appl. 436 (2012), 742-746. (2012) Zbl1231.15008MR2854904DOI10.1016/j.laa.2011.07.044

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