Rings consisting entirely of certain elements

Huanyin Chen; Marjan Sheibani; Nahid Ashrafi

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 2, page 553-558
  • ISSN: 0011-4642

Abstract

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We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ℤ 3 ⊕ ℤ 3 ; ℤ 3 ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M 2 ( ℤ 2 ) or M 2 ( ℤ 3 ) ; or the ring of a Morita context with zero pairings where the underlying rings are ℤ 2 or ℤ 3 .

How to cite

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Chen, Huanyin, Sheibani, Marjan, and Ashrafi, Nahid. "Rings consisting entirely of certain elements." Czechoslovak Mathematical Journal 68.2 (2018): 553-558. <http://eudml.org/doc/294309>.

@article{Chen2018,
abstract = {We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; $\mathbb \{Z\}_3\oplus \{\mathbb \{Z\}\}_3$; $\mathbb \{Z\}_3\oplus B$ where $B$ is a Boolean ring; local ring with nil Jacobson radical; $M_2(\mathbb \{Z\}_2)$ or $M_2(\mathbb \{Z\}_3)$; or the ring of a Morita context with zero pairings where the underlying rings are $\mathbb \{Z\}_2$ or $\mathbb \{Z\}_3$.},
author = {Chen, Huanyin, Sheibani, Marjan, Ashrafi, Nahid},
journal = {Czechoslovak Mathematical Journal},
keywords = {idempotent; nilpotent; Boolean ring; local ring; Morita context},
language = {eng},
number = {2},
pages = {553-558},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Rings consisting entirely of certain elements},
url = {http://eudml.org/doc/294309},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Chen, Huanyin
AU - Sheibani, Marjan
AU - Ashrafi, Nahid
TI - Rings consisting entirely of certain elements
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 2
SP - 553
EP - 558
AB - We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; $\mathbb {Z}_3\oplus {\mathbb {Z}}_3$; $\mathbb {Z}_3\oplus B$ where $B$ is a Boolean ring; local ring with nil Jacobson radical; $M_2(\mathbb {Z}_2)$ or $M_2(\mathbb {Z}_3)$; or the ring of a Morita context with zero pairings where the underlying rings are $\mathbb {Z}_2$ or $\mathbb {Z}_3$.
LA - eng
KW - idempotent; nilpotent; Boolean ring; local ring; Morita context
UR - http://eudml.org/doc/294309
ER -

References

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  1. Ahn, M. S., Anderson, D. D., 10.1216/rmjm/1181069429, Rocky Mountain J. Math. 36 (2006), 783-798. (2006) Zbl1131.13301MR2254362DOI10.1216/rmjm/1181069429
  2. Anderson, D. D., Camillo, V. P., 10.1081/agb-120004490, Comm. Algebra 30 (2002), 3327-3336. (2002) Zbl1083.13501MR1914999DOI10.1081/agb-120004490
  3. Breaz, S., Gălugăreanu, G., Danchev, P., Micu, T., 10.1016/j.laa.2013.08.027, Linear Algebra Appl. 439 (2013), 3115-3119. (2013) Zbl1355.16023MR3116417DOI10.1016/j.laa.2013.08.027
  4. Chen, H., Rings Related Stable Range Conditions, Series in Algebra 11. World Scientific, Hackensack (2011). (2011) Zbl1245.16002MR2752904
  5. Danchev, P. V., McGovern, W. W., 10.1016/j.jalgebra.2014.12.003, J. Algebra 425 (2015), 410-422. (2015) Zbl1316.16028MR3295991DOI10.1016/j.jalgebra.2014.12.003
  6. 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
  7. Du, X., 10.1081/AGB-120013336, Commun. Algebra 30 (2002), 4507-4525. (2002) Zbl1030.16012MR1936488DOI10.1081/AGB-120013336
  8. Immormino, N. A., Clean Rings & Clean Group Rings, Ph.D. Thesis, Bowling Green State University, Bowling Green (2013). (2013) MR3321928
  9. Kosan, M. T., Lee, T. K., Zhou, Y., 10.1016/j.laa.2014.02.047, Linear Algebra Appl. 450 (2014), 7-12. (2014) Zbl1303.15016MR3192466DOI10.1016/j.laa.2014.02.047
  10. McGovern, W., Raja, S., Sharp, A., 10.1142/S0219498815500942, J. Algebra Appl. 14 (2015). (2015) Zbl1325.16024MR3338090DOI10.1142/S0219498815500942
  11. Nicholson, W. K., 10.1007/BF01838190, Aequations Math. 9 (1973), 64-70. (1973) Zbl0255.16006MR0316497DOI10.1007/BF01838190

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