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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